  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. \'E. 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.
