 Gaussian rotation of independent copies

If $X,Y$ are independent and identically distributed random variables with a centered multivariate normal distribution and covariance matrix $\Sigma$, then $(U_{\theta },V_{\theta })$ has the same probability law as $(X,Y)$. Its two marginal covariance matrices are $\Sigma$ and its cross-covariance is zero. The pair has a multivariate normal distribution, so the two components are independent. This remains valid when $\Sigma$ is singular.