I was told that there can't exist a matrix $M\in M_3(\mathbb R)$ such that $M^TCM$ and $M^TDM$ are both diagonal,where $$C=\begin{pmatrix}1&0&0\\0&-1&0\\ 0&0&0\end{pmatrix}$$ and $$D=\begin{pmatrix}0&1&0\\1&0&0\\0&0&0\end{pmatrix}.$$ Why is this necessarily true? Can this be shown without actually writing out $$M=\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}$$ and dealing with messy algebra? I am quite sure there is a simpler way...
Added: where the resulting matrices are not the zero matrix
Thanks.