Let $G \in \mathbb{R}^{3x3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3x3}$ a symmetric and positive definite matrix. Now, the following inequality shall be proved:
$$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}) $$
If $U$ and $G$ have the same eigenvectors, both sides of the inequality are obviously equal. However for more general cases, 
I have tried to rearrange the inequality to
$$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$ 
and then using the Cauchy-Schwarz inequality in order to achieve an upper boundary that possibly fulfils the inequality. Unfortunately, I have not found a solution yet.

Thanks for any advice!