Since is an involution (i.e. it squares to the identity) one can decompose into positive and negative eigenspaces of where Since is an automorphism it follows that: where the bracketed superscripts are read modulo 2. This gives the structure of a Z2-graded algebra. The subspace forms a subalgebra of , called the even subalgebra. The subspace is called the odd part of (it is not a subalgebra). This Z2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism is called the main involution or grade involution. Elements that are pure in this Z2-grading are simply said to be even or odd. Remark. In characteristic not 2 the underlying vector space of inherits an N-grading and a Z-grading from the canonical isomorphism with the underlying vector space of the exterior algebra . It is important to note, however, that this is a vector space grading only. That is, Clifford multiplication does not respect the N-grading or Z-grading, only the Z2-grading: for instance if , then , but , not in . Happily, the gradings are related in the natural way: . Further, the Clifford algebra is Z-filtered: The degree of a Clifford number usually refers to the degree in the N-grading. The even subalgebra of a Clifford algebra is itself isomorphic to a Clifford algebra. If is the orthogonal direct sum of a vector of nonzero norm and a subspace , then is isomorphic to , where is the form restricted to and multiplied by . In particular over the reals this implies that: In the negative-definite case this gives an inclusion , which extends the sequence Likewise, in the complex case, one can show that the even subalgebra of is isomorphic to . Antiautomorphisms In addition to the automorphism , there are two antiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that the tensor algebra comes with an antiautomorphism that reverses the order in all products of vectors: Since the ideal is invariant under this reversal, this operation descends to an antiautomorphism of called the transpose or reversal operation, denoted by . The transpose is an antiautomorphism: . The transpose operation makes no use of the Z2-grading so we define a second antiautomorphism by composing and the transpose. We call this operation Clifford conjugation denoted Of the two antiautomorphisms, the transpose is the more fundamental. Note that all of these