Quadratic Wasserstein distance between Gaussian laws revisited with correlation

In this note, we give a simple derivation of the formula obtained in Dowson and Landau (1982), Olkin and Pukelsheim (1982) and Givens and Shortt (1984) for the quadratic Wasserstein distance between two Gaussian distributions on $\R^d$ with respective covariance matrices $\Sigma_\mu$ and $\Sigma_\nu$. This derivation relies on the existence of an orthogonal matrix $O$ such that $O^*\Sigma_\mu O$ and $O^*\Sigma_\nu O$ share the same correlation matrix and on the simplicity of optimal couplings in the case with the same correlation matrix and therefore the same copula.