Is it permissible in Islamic law for a brother to sign a document claiming that his brother owes him a sum of money for his previous study expenses, in order to use this document in a divorce case to reduce what his wife might demand from him in terms of money and property according to Western law, knowing that the money was previously sent to him as family assistance and not as a debt?

The basic ruling concerning lying is that it is forbidden. However, if lying is the only means to escape injustice, then it is permissible. Forcing a man to pay maintenance for a irrevocably divorced woman (talaq (definition) bain) or one whose waiting period (iddah (definition)) has ended is an injustice and a contravention of Allah's حكم (ruling/judgment). Therefore, if there is no way for your brother to be rid of this unjust law except by writing the aforementioned paper, there is no harm in it.