Pairwise Uncorrelated Jointly Gaussian Random Variables are Independent

Let $(\Omega,\mathcal{F},P)$ be a probability space, let $p$ be a natural number, and let $(X_1,\dots,X_p)$ be a Gaussian random vector on $(\Omega,\mathcal{F},P)$ whose distinct components are pairwise uncorrelated: with the covariance of square-integrable random variables, defined and finite by Square-Integrability, Moments, and Covariance Matrix of a Gaussian Random Vector,

Then the random variables $X_1,\dots,X_p$ are independent.