Pure spinors and a construction of the $E_{\ast }$-Lie algebras

Abstract

Let $(V,g)$ be a $2n$-dimensional hyperbolic space and $C(V,g)$ its Clifford algebra. $C(V,g)$ has a $\mathbb{Z}$-grading, $C^k$, and an algebra isomorphism $C(V,g)\cong End(S)$, $S$ the space of spinors. É. Cartan defined operators $L_k:End(S)\to C^k$ which are involved in the definition of pure spinors. We shall give a more refined study of the operator $L_2$, in fact, obtain explicit formulae for it in terms of spinor inner products and combinatorics, as well as the matrix of it in a basis of pure spinors. Using this information we give a construction of the exceptional Lie algebras ${\mathfrak{e}}_6,{\mathfrak{e}}_7,{\mathfrak{e}}_8$ completely within the theory of Clifford algebras and spinors.