The purpose of this chapter is to recall various results on Clifford algebras and Heisenberg algebras. The results of this chapter will be used in our construction of the hypoelliptic Laplacian over a symmetric space.

This chapter is organized as follows. In section 1.1, we introduce the Clifford algebra of a vector space *V* equipped with a symmetric bilinear form *B*.

In section 1.2, we specialize the construction of the Clifford algebra to the case of *V ⊕ V**.

In section 1.3, if (*V, ω*) is a symplectic vector space, we construct the associated Heisenberg algebra.

In section 1.4, we specialize the construction of the Heisenberg algebra to the case of *V ⊕ V**.

In section 1.5, we consider the combination ...

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