Conditional independence between two variables given any conditioning subset implies block diagonal covariance matrix for multivariate Gaussian distributions

Let x=(xV)x=(xV) be a multivariate Gaussian variable with covariance matrix ΣΣ. For ii and jj in VV, we show that if the conditional covariance between xixi and xjxj given any conditioning set K⊂V∖{i,j}K⊂V∖{i,j} is equal to zero, then ΣΣ is block diagonal and ii and jj belong to two different blocks.