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! 



Norm/trace of product inequality involving skew symmetric matrices
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\*}


