A symmetric construction of the compact real form of the Lie algebra (Q1945896)

The author gives a new construction of the compact real form of the complex Lie algebra \(E_8\). The construction has some similarities with Wilson's elementary constructions of compact forms of \(G_2\), \(F_4\) and \(E_6\). The approach is similar to that of Wilson in that the Lie algebra is written as a sum of mutually orthogonal Cartan subalgebras. The Lie product is the unique bilinear product that is preserved by the action of a certain group \(G\) of shape \(2^{5+10}\mathrm{GL}_5(2)\). A multiplicative orthogonal decomposition of a Lie algebra is a decomposition as a direct sum of orthogonal Cartan subalgebras such that the product of any two of these, is contained in a third one. The complex Lie algebra \(E_8\) has a unique multiplicative orthogonal decomposition whose group of automorphisms is the above mentioned group of shape \(2^{5+10}\mathrm{GL}_5(2)\) used in the construction.