Uncorrelated Jointly Gaussian Blocks are Independent

Let $d$ and $q$ be natural numbers and let $(X_1,\dots,X_d,Y_1,\dots,Y_q)$ be a Gaussian random vector on a probability space $(\Omega,\mathcal{F},P)$ such that, with the covariance of square-integrable random variables (defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector),

that is, every $X_i$ is uncorrelated with every $Y_k$. Then the $\sigma$-algebras $\sigma(X_1,\dots,X_d)$ and $\sigma(Y_1,\dots,Y_q)$ are independent.