Mutual independence of Quadratic forms of a random vector in $\mathbf{R}^n$

Question

Assume a random vector $X \in \mathbf{R}^n$ with the distribution $N(\mu, \sigma^2 I)$. Consider the quadratic forms describe by three orthogonal perpendicular matrices $M_1, M_2, M_3$ and we want to show that the quadratic form $$X^T M_i X, \quad i = 1, 2,3. $$ are mutually independent. Clearly, we could use the fact that if $M_iM_j = 0, \forall i\neq j = 1,2,3,$ then the pairs $$X^TM_iX \text{ and } X^TM_jX, \quad \forall i\neq j $$ are pairwise independent. Now, for mutually independent, $\textbf{recall:}$ three events are said to be mutually independent if the three events are pairwise independent such that $$P(A \cap B \cap C) = P(A) P(B) P(C)$$ So, is it right to conclude that the three quadratic forms are mutually independent if they are pairwise independent such that $$M_1M_2M_3 = 0$$ I haven't yet seen any theorem about mutually independent of quadratic forms.