E7 groups from octonionic magic square

Understand

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.