Since you have -  Peter appeals to the genuineness of his readers' conversions, a radical change they were well aware of. One of the radical changes of this new birth is that it brings is love for our brethren. You do have love for your Christian brethren don't you?

In obedience to the truth - Synonymous with believing the truth. We should not separate belief and obedience for true faith is obeying faith. Those who teach you can believe the truth but not obey the truth have difficulty with passages like this.

John 3:36 He who believes (present tense = as the general direction of their life) in the Son has eternal life; but he who does not obey (disbelieves wilfully and perversely and as a lifestyle = present tense) the Son shall not see life, but the wrath of God abides on him."

See related topic - Study of phrase Obedience of faith

What is the evidence of a purified soul in this context? Love for the brethren. Holy living is incomplete if it isn’t accompanied by love. And such love is now possible for born again persons for as Paul explained to the believers in Rome...

the love of God has been poured out within our hearts through the Holy Spirit Who was given to us. (see note Romans 5:5)

Peter alluded to obedience in his opening words explaining that his readers were chosen...

according to the foreknowledge of God the Father, by the sanctifying work of the Spirit, that you may obey Jesus Christ and be sprinkled with His blood: May grace and peace be yours in fullest measure. (see note 1 Peter 1:2)

In Romans Paul the phrase obedience of faith is like a pair of "bookends" as it were enclosing Paul's magnum opus on the gospel of Jesus Christ...

but now is manifested, and by the Scriptures of the prophets, according to the commandment of the eternal God, has been made known to all the nations, leading to obedience of faith;  (See note Romans 16:26)

John Piper comments on the critical importance of understanding the phrase obedience of faith writing...

Obedience (5218) (hupakoe  from hupo = under + akouo = hear) (Click in depth word study of hupakoe) literally means to "hear under" which conveys the picture of attentive hearkening, of listening and following instructions, of being in compliance or of listening and submitting to that which is heard.

Hupakoe conveys the picture of one listening and following instructions. Submitting to that which is heard involves a change of attitude, forsaking the tendency of the fallen nature to rebel against Divine instructions and commands and seeking God's will, not self will. Someone has said that a "proof" that we are of the elect is not an empty prating about how secure we are once we believed, but rather how sensitive we are to the principle and practice of obedience to Jesus.

Vincent notes that hupakoe was a

"peculiarly New Testament term unknown in classical Greek."

Here in 1 Peter "obedience to the truth" refers to subjection to the saving will of God revealed in Christ. It is notable that this is the second time Peter describes saving faith as an act of obedience (see note 1 Peter 1:2).

In Romans, Paul twice uses the phrase “the obedience of faith.” (Ro 1:5,16:26) We should not try to separate belief and obedience. True faith is obeying faith (see Heb 3:18-19 where you see unbelief paralleled with disobedient.)

Illustration - A missionary translator was endeavoring to find a word for “obedience” in the native language. This was a virtue seldom practiced among the people into whose language he wanted to translate the New Testament. As he returned home from the village one day, he whistled for his dog and it came running at full speed. An old man, seeing this, said, admiringly in the native tongue, “Your dog is all ear.” Immediately the missionary knew he had his word for obedience. (Encyclopedia of 7700 Illustrations by Paul Lee Tan)

Wayne Barber reiterates this important truth emphasizing that believers have a

"willingness to obey. That doesn’t mean Christians always obey. But you cannot habitually live disobedient unto God. You may have an area that will trip you up. But at some point you will confess, repent and come back to God. Why? Because Life is inside of you. It is a Person. The divine Seed of Life is in you, and you can’t be left to do what you want to do. God will either take you out of here, or He’ll prune you. He’ll cut you back and cut you back and cut you back until finally He disciplines. He scourges and chastens those whom He loves. He doesn’t let us get away with lawless living. You find a person who claims to know Christ and lives in sin, lives lawlessly as a habitual practice, that person does not know Christ" (see 1Jn 2:10-11, 3:9-10, 5:2).

You...purified - Obviously we do not have the power to produce personal purity, but God it is the Holy Spirit Who purifies our souls when we are saved.

Purified (48) (hagnizo from hagnos = pure from contamination) means to make clean and was used to describe external ceremonial purification (in both the Septuagint =LXX and the NT) but in the present context describes an internal, supernatural cleansing which occurred when they received the living and abiding word and were caused to be born again by God (see discussion of 1 Peter 1:3).

Hagnizo is in the perfect tense signifying a past act (the moment we believed the gospel) with ongoing effects (enabled to sacrificially love others). This tense underscores the permanence of the once for all transaction of salvation. The point is that one's salvation cannot be lost. Even the verb tense (perfect) supports the eternal security of the believer and counters the tragic false teaching to the contrary.

Matthew Henry comments that...

To purify the soul supposes some great uncleanness and defilement which had polluted it, and that this defilement is removed. Neither the Levitical purifications under the law, nor the hypocritical purifications of the outward man, can effect this. The word of God is the great instrument of a sinner’s purification. The gospel is called truth, in opposition to types and shadows, to error and falsehood. This truth is effectual to purify the soul, if it be obeyed, John 17:17. Many hear the truth, but are never purified by it, because they will not submit to it nor obey it...The souls of Christians must be purified before they can so much as love one another unfeigned. There are such lusts and partialities in man’s nature that without divine grace we can neither love God nor one another as we ought to do; there is no charity but out of a pure heart. It is the duty of all Christians sincerely and fervently to love one another. Our affection to one another must be sincere and real, and it must be fervent, constant, and extensive.

Our position (purified) forms the basis for our practice (fervent love). Note that it is God of course Who purifies our souls when we are saved for fallen men and women do not have the power to bring about personal internal moral purity.

FOR A SINCERE LOVE OF THE BRETHREN: eis philadelphian anupokriton: (1 Peter 2:17; 3:8; 4:8; John 13:34,35; 15:17; Romans 12:9,10; 2 Corinthians 6:6; Ephesians 4:3; 1 Thessalonians 4:8,9; 1 Timothy 1:5; Hebrews 6:10; 13:1; James 2:15,16; 2 Peter 1:7; 1 John 3:11,14-19,23; 4:7,12,20,21; 5:1,2 )

a fraternal affection without hypocrisy (Clarke)

For a sincere love - literally unto or into a sincere love. And so one of the primary goals of our salvation is that we might show love to fellow believers.

Why is demonstration of love so important? For one thing John writes...

We know that we have passed out of death into life, because we love the brethren. He who does not love abides in death. (1 Jn. 3:14)

And in John's gospel Jesus explains to His disciples...

A new commandment I give to you, that you love one another, even as I have loved you, that you also love one another. By this all men will know that you are My disciples, if you have love for one another.  (John 13:34-35).

Sincere (505) (anupokritos from "a" = without +  hupokrinomai = to pretend) means unfeigned, without pretense, that is that which is genuine, free from deceit, authentic, undisguised, without pretense or sham and to use the "king's old English" to be "without dissimulation" (KJV) (dissimulate = hide under a false appearance).

In classical Greek drama, the hupokrites (actor) wore a face-mask projecting an image but hiding his true identity under (hupo) a mask.

Peter is saying that the Christian’s love should not be acting a part or wearing a mask, but should be an authentic expression of goodwill.

In a similar way Paul exhorts the believers at Rome to...

Let love be without hypocrisy (anupokritos). Abhor what is evil; cling to what is good.  (See note Romans 12:9)

Believers are not simply to be "playing the part" but are to exemplify a love which is completely genuine, unhypocritical, without pretense or deceit. A believer's love is to be "the real thing".

Wuest comments that

The world wears a mask. The love which it shows on the face is only external. That is feigned love. Ours should be unfeigned. If a saint does not have a love which is unfeigned, the trouble is with his adjustment to the Holy Spirit who is the One to provide that saint with that love. The Spirit-filled saint does not have to play the hypocrite in the matter of love, for love shines right out of his eyes. It is on his face, in his actions." (Wuest's word studies from the Greek New Testament: Grand Rapids: Eerdmans)

Anupokritos is used by the NT writers to modify

love (Romans 12:9, 2Cor 6:6, 1 Peter 1:22),

faith (present verse and 1Ti 1:5) and

wisdom (James 3:17).

Metaphorically and morally, a hupokrites (a hypocrite) is anyone who pretends to be something they are not. It is interesting to note that our English word sincere comes from the Latin sincerus, which means "without wax" which stems from a practice of the early Roman merchants who set their earthen and porcelain jars out for sale. If a crack appeared in one, they would fill it with wax the same color as the jar, so a buyer would not be aware that it was cracked. But astute buyers learned to hold these jars out in the sun, and if the jar was cracked, the wax would melt and the crack would be revealed. So the honest merchants would test their wares this way and mark them sincerus -- "without wax".

Hypocrisy is exceeded in evil only by unbelief. The consummate hypocrite in Scripture, Judas, was also the consummate egoist. He feigned devotion to Jesus to achieve his own selfish purposes. His hypocrisy was unmasked and his self-centeredness was made evident when he betrayed Jesus for the thirty pieces of silver. Peter's exhortation for us to manifest an unfeigned love implies that there can be a feigned "love". Don't be deceived. It is tragic when people try to “manufacture” love, because the product is obviously cheap and artificial. The love that we share with each other, and with a lost world, must be generated by the Spirit of God. It is a constant power in our lives, and not something that we turn on and off like a radio.

John Calvin says

nothing is more difficult than to love our neighbors in sincerity. For the love of ourselves rules, which is full of hypocrisy; and besides, every one measures his love, which he shows to others, by his own advantage, and not by the rule of doing good. He adds, fervently; for the more slothful we are by nature, the more ought every one to stimulate himself to fervor and earnestness, and that not only once, but more and more daily.

By nature, all of us are selfish and it therefore took a work of grace to supernaturally give believers the selfless, sacrificial kind of love that God is and that He displays undeserving sinners. Because we “obeyed the truth through the Spirit,” God purified our souls and poured His love into our hearts, Paul recording that...

hope does not disappoint, because the love of God has been poured out within our hearts through the Holy Spirit who was given to us. (see note Romans 5:5).

Love of the brethren  (5360) (philadelphia from phílos = beloved, dear, friendly + adelphós = brother) means "fraternal love" or brotherly love (kindness). Brotherly love normally referred to the love members of a family held for each other (this was the way it was used in secular Greek) and would not normally be used to describe the love between members of different families. However, in the NT philadelphia is used to describe the love that believers possess for one to another, for even though they were members of different natural families, they were united in Christ and were recipients of family love originating from the Father Who had bestowed His great love on His spiritual children...

See how great a love the Father has bestowed upon us, that we should be called children of God; and such we are. For this reason the world does not know us, because it did not know Him (1John 3:1)

and in your godliness, brotherly kindness, and in your brotherly kindness, love. (See note 2 Peter 1:7)

Love for the brethren is an evidence that we truly have been born of God, as John explains in his first epistle...

Beloved, let us love one another, for love is from God; and everyone who loves is born of God and knows God. 8 The one who does not love does not know God, for God is love. 9 By this the love of God was manifested in us, that God has sent His only begotten Son into the world so that we might live through Him. 10 In this is love, not that we loved God, but that He loved us and sent His Son to be the propitiation for our sins. 11 Beloved, if God so loved us, we also ought to love one another. 12 No one has beheld God at any time; if we love one another, God abides in us, and His love is perfected in us. (1Jn 4:7-21).

Now as obedient children (see note 1 Peter 1:14) we are no longer to live in the selfish desires of our old nature, for believers are partakers of His divine nature and have a new to love.

FERVENTLY LOVE ONE ANOTHER FROM THE HEART: ek (katharas) kardias allelous agaphesate (2PAAM) ektenos: (Philippians 1:9; 1 Thessalonians 3:12; 2 Thessalonians 1:3; Revelation 2:4)

KJV (Textus Receptus) qualifies the NAS from the heart, but adding

love one another with a pure heart fervently

From the heart (or with a pure heart) - When we entered the New Covenant by grace through faith, we received a new heart and God's Spirit (see Ezekiel 36:26-27 which is an OT promise of the New Covenant). Now with our new heart, enabled by the Spirit, our motive in loving one another is not to get but to give. The world teaches that if you give to others, you will be able to manipulate them and ultimate fulfill your selfish desires. Love from the (pure) heart never seeks to use others to its advantage.

Because of our new position (purified souls) we are commanded to a new practice (sincere love of the brethren). In other words, our position in Christ forms the basis for our practice in the power of His Spirit (cf notes Galatians 5:22;  Galatians 5:23). The principle that doctrine determines duty permeates the Scriptures, for as James said...

if anyone is a hearer of the word and not a doer, he is like a man who looks at his natural face in a mirror for once he has looked at himself and gone away, he has immediately forgotten what kind of person he was. (James 1:23-24)

Fervently love one another - Matthew Henry writes that this exhortation...

supposes that the gospel had already had such an effect upon them as to purify their souls while they obeyed it through the Spirit, and that it had produced at least an unfeigned love of the brethren; and thence he argues with them to proceed to a higher degree of affection, to love one another with a pure heart fervently

Paul's prayer for the saints at Philippi was to manifest and experience an abounding love for one another...

And this I pray, that your love may abound still more and more in real knowledge and all discernment, so that you may approve the things that are excellent, in order to be sincere and blameless until the day of Christ (see notes Philippians 1:9; 1:10)

Paul gives a similar charge to the saints in Thessalonica who have come out of paganism and idol worship exhorting them...

Now as to the love of the brethren, you have no need for anyone to write to you, for you yourselves are taught by God to love one another; for indeed you do practice it toward all the brethren who are in all Macedonia. But we urge you, brethren, to excel still more. (See note 1Thessalonians 4:9;  4:10)

Fervently  (1619) (ektenos from ek = out + teíno = stretch; English = tension, etc) literally pictures one "stretching out" to love others! It pictures "an intense strain" and unceasing activity which normally involving a degree of intensity and/or perseverance. 

Ektenos is from the verb ekteino which means to “stretch out the hand.”, thus it means to be stretched out—earnest, resolute, intense. The fundamental idea is earnestness, zealousness -- not doing something lightly and perfunctorily but straining as it were to do it!

Ektenos was an athletic term conveying the meaning of “striving with all of one’s energy” and was used to describe a runner who was moving at maximum output with taut muscles straining and stretching to the limit. This meaning presents the clear picture that love is not something that will just happen, but is something we have to work at like an Olympic athlete who strives to master his area of expertise with all his energy. We must make the choice and be earnest, resolute, even intense in our practice of agape love, always in complete dependence of God's indwelling Spirit and His living and abiding Word.

Peter is saying in essence

Stretch to the limits in your loving others sacrificial and selflessly.

Roger Raymer adds that...

This love is to be expressed not shallowly but “deeply” (ektenos, “at full stretch” or “in an all-out manner, with an intense strain”...). (Walvoord, J. F., Zuck, R. B., et al: The Bible Knowledge Commentary. 1985. Victor or Logos)

Vincent writes that ektenos is a compound work

with the verb teino, to stretch, and (signifies) intense strain; feeling on the rack (an instrument of torture on which a body is stretched)

Vine says...

The idea suggested is that of not relaxing in effort, or acting in a right spirit.

In the only other NT use, ektenos describes the church's prayer for Peter in prison

So Peter was kept in the prison, but prayer for him was being made fervently by the church to God. ( Acts 12:5) (Note that the Textus Receptus - used to translate the KJV - actually has the closely related word ektenes not ektenos as does the Nestle-Aland which is used for the NAS translation.)

The comparative form of the closely related adjective ektenes (ektenesteron) is used to describe the intensity of our Lord's prayer in Gethsemane

And being in agony He was praying very fervently (ektenesteron); and His sweat became like drops of blood, falling down upon the ground. (Luke 22:44) (Comment: what a picture of the passion of Jesus in prayer!)

Note that ektenos is used only here and in some manuscripts (Nestle-Aland) in Acts 12:5. As mentioned above, Acts 12:5 in the Textus Receptus uses the closely related adjective ektenes., which is also used in chapter 4 of first Peter...

Above all, keep fervent (ektenes) in your love for one another, because love covers a multitude of sins. (See note 1 Peter 4:8)

Fervently love one another - MacDonald comments that...

The exhortation to love one another is especially timely for a people undergoing persecution because it is well known that “under conditions of hardship, trivial disagreements take on gigantic proportions.” (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson or Logos)

Love  (25) (agapao) (Click study of agapao) speaks of an unconditionally, sacrificial love, which is ultimately the love that God demonstrates and even "is" ("God is love" 1 John 4:8, 16). By nature, in Adam (see note Romans 5:12), all men are are selfish and therefore only a a miracle of God could give sinners this quality of divine, saintly love.

Agape love is neither a not sentimental or emotional type love so often depicted in television, movies and magazines. Instead, agape love demands an act of obedience by the lover for the "lovee" and thus represents a specific volitional choice or act of the lover's will.

Agape love desires the recipients highest good whether one they deserve it or not or whether the lover feels like giving it or not. If is not a matter of how one feels. Feelings can be deceiving and can emanate from our fallen flesh nature. Too many marriages are falling apart because one party doesn't feel like they love the other party any longer. That is unbiblical for a believer, for we are commanded to demonstrate a love based on God's truth and we can carry out this command empowered by God's Spirit as we yield to His control in every circumstance that might cause us to not "feel" like loving the other person.

Agape love is not conditional but is to be given to one's spouse or any other person, believer or not even if or when it is not received or is not returned! "Impossible" you protest! Indeed, this supernatural love is not humanly possible but only possible because God's Spirit is within us both to give us the "want to" and to give the "power to" carry it out. (See notes Philippians 2:12; 2:13)

Note that in this verse, the verb "love" is not a suggestion but a clear command to be immediately (even urgently) obeyed each time one encounters a person or circumstance that might otherwise tempt us to rely on and respond out of our fallen nature. To grow in grace is to recognize those circumstances (especially "adverse" circumstances) and those people (especially "difficult" people) and then surrendered to and empowered by the Holy Spirit (in other words as you encounter these circumstances or people, you are already walking in the Spirit, being controlled by the Spirit) the giving of agape love is your "supernatural reflex". In other words you act (based on the truth) don't react (based on your feelings)!

The aorist tense and imperative mood  commands the believer to carry out this act of love now and do it effectively. The