Theorem: Let $X \in \mathbb{R}^d$ be a Gaussian vector and $\Gamma_X$ its covariance matrix. Then $X_1,\dots,X_d$ are independent if and only if $Γ_X$ is a diagonal matrix (i.e. the variables are pairwise uncorrelated)
My attempt: One direction is easy: If they are pairwise uncorrelated then the non-diagonal entries of $\Gamma_X$ would equal $0$, being covariances of independent variables.
What troubles me is the other direction - I though about starting with a vector of $d$ independent standard normal variables and then scaling their variances by pre-multiplying the vector by a matrix and hence preserving the property of being a Gaussian vector. (Notice that WLOG we can assume the vector to have entries with mean $0$) However this doesn't look very rigorous since I end up with $2$ vectors giving the same covariance matrix with only one of them having independent entries.