Fourth order statistc of sign function of random vector

Question

we have this preposition as follows

$\textbf{Proposition}$: $\textit{let} \ \boldsymbol{\zeta} \sim \mathcal{N}(\mathbf{0},{\gamma}\mathbf{I}_N). For \ \mathbf{a}_1 , \mathbf{a}_2 \in \ \mathbb{R}^{N \times 1}$

$\mathbb{E} [sgn({\mathbf{a}_1}^T \boldsymbol\zeta) sgn({\mathbf{a}_2}^T \boldsymbol\zeta)] = \Omega \bigg(\frac{{\mathbf{a}_1}^T \mathbf{a}_2}{||\mathbf{a}_1||||\mathbf{a}_2||} \bigg)$

where sgn is the sign function. $\Omega(x) = \frac{2}{\pi}$ arcsine($x$).

My question is how we can find following expression in closed-form

$\mathbb{E} [sgn({\mathbf{a}_1}^T \boldsymbol\zeta) sgn({\mathbf{a}_2}^T \boldsymbol\zeta)\ \mathbf{A}\ sgn({\mathbf{b}_1}^T \boldsymbol\eta) sgn({\mathbf{b}_2}^T \boldsymbol\eta)] $

where matrix $\mathbf{A}$ is given, and the definition of $\mathbf{b}$ and random vector $\boldsymbol\eta$ is the same with $\mathbf{a}$ and $\boldsymbol\zeta$ respectively, just with different size (lets say $\in \ \mathbb{R}^{M \times 1}$). Indeed, I'm looking for another preposition for this case.

thanks