Extension of Pauli's Theorem to Clifford Algebras

Analogues of Pauli's theorem are proved for arbitrary two sets consisting of an even or odd number of elements satisfying the defining relations of the Clifford algebra. Given two sets of square complex matrices of some order, there exists an unique matrix. The vector subspaces spanned by the elements are indexed by ordered multi-indices of length k. For the real Clifford algebra of odd dimension, the corresponding sets generate a basis of the Clifford algebra or take the values ∓e and then the sets do not generate a basis.