Norm/trace of product inequality involving skew symmetric matrices

Question

I wonder if the following inequality involving skew symmetric matrices is true:

Suppose that $B,C \in \mathbb{R}^{d \times d}$ are skew-symmetric matrices, and $\Sigma \in \mathbb{R}^{d \times d}$ is positive-definite. Then,

$$\mbox{Tr}(B^2 \Sigma C^2) - \frac{1}{2}\mbox{Tr}((CB) \Sigma (CB) + (CB) \Sigma (BC)) \geq 0 $$

For $\Sigma = I_d$, this is a relatively well-known inequality due to Bellman (and also follows from Araki-Lieb-Thirring). Not sure what can be said for $\Sigma$ beyond identity.

Any relevant tools/inequalities appreciated!

Answer 1

Something seems to be missing here, because the inequality is trivially seen to be false. Consider the following randomly picked matrices for instance:

\begin{equation*} B = \begin{bmatrix}0 & -4 & 4\\ 4 & 0 & -10\\ -4& 10 & 0\end{bmatrix},\ C = \begin{bmatrix}0 & -6 & 11\\ 6 & 0 & -12\\ -11& 12 & 0\end{bmatrix},\ \Sigma = \begin{bmatrix}21 & 22 & -2\\ 22 & 53 & 3\\ -2& 3 & 21\end{bmatrix}. \end{equation*}