Let be a finite-dimensional real or complex vector space with a nondegenerate quadratic form . The (real or complex) linear maps preserving form the orthogonal group . The identity component of the group is called the special orthogonal group . (For real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, has a unique connected double cover, the spin group . There is thus a group homomorphism whose kernel has two elements denoted , where is the identity element. Thus, the group elements and of are equivalent after the homomorphism to ; that is, for any in .