If we have three random variables $X,Y,Z$, which are jointly normal, how can it be shown that $I(X;Y) = I(X;Z)= 0 \implies I(X;Y;Z) = 0$?
I know that for jointly normal distributions $X,Y,Z$: $I(X;Y) = \frac{1}{2} \log \frac{\det(\Sigma_X) \det(\Sigma_Y)}{\det(\Sigma_{XY})}$. Knowing this much, how can I move on from here?
And how is this not true for the general case of jointly (non-normal) distributed random variables $X,Y,Z$?