It is well-known that every square matrix can be written as the sum of a symetric matrix and an antisymetrix matrix.
The same does not hold for the product : for example, matrices $M\in\mathscr{M}_n(\mathbb{K})$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$) with $\text{tr}(M)\neq 0$ or non singular $M$ with odd $n$ cannot be written as $S\times A$ with $S\in\mathscr{S}_n(\mathbb{K})$ and $A\in\mathscr{A}_n(\mathbb{K})$.
One easily checks that if $n=2$ and $M\in\mathscr{M}_n(\mathbb{K})$ has zero trace, then it can be written as $S\times A$ with $S\in\mathscr{S}_n(\mathbb{K})$ and $A\in\mathscr{A}_n(\mathbb{K})$.
My question is : does this result hold for even $n=4,6,8,...$ ?