Issue: I have two random column vectors $a,b\in\mathbb{R}^L$ and would like to simplify the following expression: $$ \begin{bmatrix} \mathbb{E}[aa^\top] & \mathbb{E}[bb^\top] \end{bmatrix} \begin{bmatrix} \mathbb{E}[aa^\top] & \mathbb{E}[ab^\top] \\ \mathbb{E}[ba^\top] & \mathbb{E}[bb^\top] \end{bmatrix}^{-1}. $$
Simplify here means I would like to rewrite the matrix such that I do as little inversion as possible and also invert as small a matrix as possible.
Attempt: I tried using all the various block matrix inversion formulas, but I couldn't simplify the expression after substituting the various formulas. Is there another way or is this as simple as we can go? Thanks.