Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors,
and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$.
Can we write the distribution of $$\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$$ as a function of $u$?
That is, if we randomly project two points onto normal distributed (unit) vectors, I wonder how the inner product is distributed? We know that for non-normalized $A$ and $B$ the distance is nicely distributed after projection (per 2-stability), but I can't seem to find strong arguments for or against a similar property holding for inner products.