Let C be a Clifford algebra of a quadratic space V and suppose that e1, e2, ..., en is a basis for V. Let C_ + be the subspace of linear combinations of products of an even number of basis elements and let C_ - be the subspace of linear combinations of products of an odd number of basis elements. Then C_ + is a subalgebra and C is the direct sum of C_ + and C_ -. The main involution of C is the automorphism J such that J(u) = u if u∈C_ + and J(u) = - u if u∈C_ -.
The main involution of the Clifford algebra C.
The mapping C to C which reverses the multiplication is an antiautomorphism whose square is the identity; it is called the main antiautomorphism of C.
The main antiautomorphism of C. The first time this function is invoked it sets the attribute antiAutMat of C to the matrix defining this map.