Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $\begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix

$$ \begin{bmatrix} A & B & 0\\ B^{\top} & C & -B^{\top} \\ - A & -B & A\end{bmatrix} $$ has eigenvalues with positive real part. Numerical tests suggest this is true, but I cannot prove it.