A quaternionic construction of 𝐸₇

Abstract

We give an explicit construction of the simply-connected compact real form of the Lie group of type E 7 E_7 , as a group of 28 × 28 28\times 28 matrices over quaternions, acting on a 28 28 -dimensional left quaternion vector space. This leads to a description of the simply-connected split real form, acting on a 56 56 -dimensional real vector space, and thence to the finite quasi-simple groups of type E 7 E_7 . The sign problems usually associated with constructing exceptional Lie groups are almost entirely absent from this approach.