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For every nonzero $v\in {\mathbb{R}}^{n-p}$, full column rank of $L$ gives $Lv\ne 0$, and henceThus the Gram matrix $L^{T}L$ is a positive-definite matrix and is invertible. Using $BA=0$ and $A=L{L}^T$,so

A linear image of a multivariate normal vector is again multivariate normal. Therefore the stacked vector has mean $(B\mu ,L^T\mu {)}^T$ and covarianceIn particular,The two diagonal blocks are positive-definite by the rank assumptions. The two blocks are jointly Gaussian and have zero cross covariance. The fact that uncorrelated jointly Gaussian variables are independent makes $BY$ and $L^{T}Y$ independent. Sinceis a measurable function of the second block, independence is preserved under this transformation. This proves Gaussian independence of linear and quadratic statistics: