If a variable $x$ is normally distributed with mean zero, we have previously given a necessary and sufficient condition (see references at end of this paper) for the independence of two real symmetric quadratic forms in $n$ independent values of that variable. This condition is that the product of the matrices of the forms should vanish. In the present paper, we have proved that the same algebraic condition is both necessary and sufficient for the independence of two real symmetric bilinear, or a real symmetric bilinear and quadratic form, in normally correlated variables.
"Bilinear Forms in Normally Correlated Variables." Ann. Math. Statist. 18 (4) 565 - 573, December, 1947. https://doi.org/10.1214/aoms/1177730347