Is it permissible to sell a piece of gold to the financially distressed brother without the husband's knowledge, on the condition that the brother repays the value of the gold later?

If the gold is entirely your property, it is permissible to sell it and lend its price to your brother without your husband's knowledge, because a mature woman has the right to dispose of her wealth without her husband's permission. However, selling the gold itself to him on a deferred payment basis is not permissible.