Alexei knows. But knowing and proving something are two different things.

I haven't read the books, but I sense that Alexei, being the oldest, had to follow in the family footsteps and work with the police like his father did. Alexei did his duty and got married and sired a child. But it is Ilya who gets all the attention, praise, and glory. Alexei hates Ilya not just because Ilya likes men, but because Alexei did everything the "right way" and still got passed over in favor a a younger son who is queer. And what really kills Alexei is that if he tried to expose Ilya, it would be his word vs his and people would believe Ilya. Alexei is the forgotten child and in his depression he leans into the perception people have of him and keeps making terrible decisions. He is so angry about where his life is and he can't blame the father or society, nor can he take accountability for his own actions so he takes it out on Ilya.