isomorphic to the tensor product of the Clifford algebras of and , which is the space with its quadratic form multiplied by . Over the reals, this implies in particular that These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see the classification of Clifford algebras. Notably, the Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends only on the signature . This is an algebraic form of Bott periodicity. Lipschitz group The class of Lipschitz groups ( Clifford groups or Clifford–Lipschitz groups) was discovered by Rudolf Lipschitz. In this section we assume that is finite-dimensional and the quadratic form is nondegenerate. An action on the elements of a Clifford algebra by its group of units may be defined in terms of a twisted conjugation: twisted conjugation by maps , where is the main involution defined above. The Lipschitz group is defined to be the set of invertible elements that stabilize the set of vectors under this action, meaning that for all in we have: This formula also defines an action of the Lipschitz group on the vector space V that preserves the quadratic form Q, and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elements r of V for which Q(r) is invertible in K, and these act on V by the corresponding reflections that take v to . (In characteristic these are called orthogonal transvections rather than reflections.) If V is a finite-dimensional real vector space with a non-degenerate quadratic form then the Lipschitz group maps onto the orthogonal group of V with respect to the form (by the Cartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm. Spinor norm In arbitrary characteristic, the spinor norm Q is defined on the Lipschitz group by It is a homomorphism from the Lipschitz group to the group K× of non-zero elements of K. It coincides with the quadratic form Q of V when V is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from