  For any manifold M, the direct sum TM \oplus T*M carries a natural inner
product given by the pairing of vectors and covectors. Differential forms on M
may be viewed as spinors for the corresponding Clifford bundle, and in
particular there is a notion of \emph{pure spinor}.
  In this paper, we study pure spinors and Dirac structures in the case when
M=G is a Lie group with a bi-invariant pseudo-Riemannian metric, e.g. G
semi-simple. The applications of our theory include the construction of
distinguished volume forms on conjugacy classes in G, and a new approach to the
theory of quasi-Hamiltonian G-spaces.
