$E_7$ groups from octonionic magic square

Abstract

In this paper, we continue our program, started in Euler angles for $G(2)$, of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for $E_7$, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra $\mathfrak{J}$ and the quaternionic algebra $\mathbb{H}$, both in the adjoint and the 56-dimensional representations. Then, we provide the Euler parametrization of $E_7$ starting from its maximal subgroup $U = (E_6 \times U(1))/\mathbb{Z}_3$. Next, we give the constructions for all the other maximal compact subgroups.