E7 groups from octonionic magic square

TL;DR

This paper develops explicit matrix realizations and Euler angle parameterizations for the exceptional Lie group E7, extending previous work on exceptional groups and utilizing the octonionic magic square construction.

Contribution

It provides the first explicit matrix realizations and Euler parametrizations of E7 based on the Tits construction and maximal subgroup decompositions.

Findings

Explicit matrix realizations of E7 via Tits construction

Euler parametrization of E7 from its maximal subgroup

Construction of all maximal compact subgroups of E7

Abstract

In this paper we continue our program, started in [2], of building up explicit generalized Euler angle parameterizations for all exceptional compact Lie groups. Here we solve the problem for E7, by first providing explicit matrix realizations of the Tits construction of a Magic Square product between the exceptional octonionic algebra J and the quaternionic algebra H, both in the adjoint and the 56 dimensional representations. Then, we provide the Euler parametrization of E7 starting from its maximal subgroup U=(E6 x U(1))/Z3. Next, we give the constructions for all the other maximal compact subgroups.