MHB Volume of $E$ in $\mathbb{R}^4$ - AMM

[MountEvariste]

Let $E$ be the set of all real $4$-tuples $(a, b, c, d)$ such that if $x, y \in \mathbb{ R}$, then:
$(ax+by)^2+(cx+dy)^2 \le x^2+y^2$.
Find the volume of $E$ in $\mathbb{R}^4$.​

Source: AMM.

 

[MountEvariste]

Hint:

Spoiler

Show that $E$ is defined by the inequality $a^2 +b^2 +c^2 +d^2 \leqslant 1+(ad −bc)^2$ with $a^2 +c^2 \leqslant 1$ and $b^2 +d^2 \leqslant 1$.