All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
If the deceased did not leave any heirs besides those mentioned in the question, then after executing the will, the inheritance should be divided amongst his two sons equally by Ta’seeb (by virtue of having a paternal relation with the deceased and not having an allotted share, so they get what is left after the allotted shares have been distributed). His two full-brothers do not get anything from the inheritance as they are excluded from it because of the existence of the sons of the deceased. Ibn Al-Munthir, said: “The scholars are in consensus that full-brothers and full-sisters and half-brothers and half-sisters from the father’s side do not inherit if there is a son or a grandson or even a great grandson. They also do not inherit if the father is alive.”
Therefore, the inheritance should be divided into two shares, and each son gets one share (after executing the will).
As regards the deceased's will of one-third for the "adopted" daughter who does not inherit him, it is a valid and effective will. So, that daughter takes one-third of the estate and the remainder will be divided equally among the two sons. However, the will should be verified by the testimony of eligible witnesses or the heirs should approve of it and attest to it. The jurists have stated the ways to verify wills; for instance Ibn Qudaamah said in Al-Kaafi: “The testimony of one man and two women is acceptable for the binding transfer of property, as Allaah says (what means): {O you who have believed, when you contract a debt for a specified term, write it down. And let a scribe write [it] between you in justice. Let no scribe refuse to write as Allaah has taught him. So let him write and let the one who has the obligation [i.e. the debtor] dictate. And let him fear Allaah, his Lord, and not leave anything out of it. But if the one who has the obligation is of limited understanding or weak or unable to dictate himself, then let his guardian dictate in justice. And bring to witness two witnesses from among your men. And if there are not two men [available], then a man and two women from those whom you accept as witnesses.}[Quran 2:282]. However, Ibn Abi Moosa said: “The will cannot be approved unless there are two male witnesses, as Allaah says (what means): {O you who have believed, testimony [should be taken] among you when death approaches one of you at the time of bequest — [that of] two just men from among you.} [Quran 5:106]" [End of quote]
It is a judge who ratifies such approval of the deceased's will.
Please refer to Fatwa 82371 about the ruling on adoption.
Allaah Knows best.