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,

$$\mbox{Tr}\left((B^2)^T \Sigma B^2\right) \mbox{Tr}\left((C^2)^T \Sigma C^2\right) \geq \left[\frac{1}{4} \mbox{Tr}\left((BC + CB)^T \Sigma (BC + CB)\right)\right]^2 $$

When $\Sigma = I_d$ it reduces to Cauchy-Schwartz after a little rearrangement of the RHS. However, if $\Sigma \neq I_d$, it doesn't reduce to a "different norm" (the one suggested by $\Sigma$) Cauchy-Schwartz.

Any relevant tools/inequalities appreciated!