 Gaussian independence of linear and quadratic statistics

When $L$ has full column rank, its Gram matrix $L^{T}L$ is positive-definite and $BL{L}^{T}L(L^{T}L{)}^{-1}=BL=0$. The stacked normal vector $(BY,L^{T}Y)$ therefore has zero cross covariance, so its blocks are independent. Since $Y^{T}AY=\Vert L^{T}Y{\Vert }^2$, the linear and quadratic statistics are independent. This is the projection argument behind independence of regression coefficient and variance estimators in a normal linear model.