(This may or may not decompose into irreducible representations.) The space of spinors may also be defined as a spin representation of the orthogonal Lie algebra. These spin representations are also characterized as the finite-dimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finite-dimensional group representation of the spin group on which the center acts non-trivially. There are essentially two frameworks for viewing the notion of a spinor. From a representation theoretic point of view, one knows beforehand that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents "spinors". From this view, a spinor must belong to a representation of the double cover of the rotation group , or more generally of a double cover of the generalized special orthogonal group on spaces with a metric signature of . These double covers are Lie groups, called the spin groups or . All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued projective representations of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.) From a geometrical point of view, one can explicitly construct the spinors and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of the spinors, such as Fierz identities, are needed. The language of Clifford algebras (sometimes called geometric algebras) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras. It largely removes the need for "ad hoc" constructions. In detail, let "V" be a finite-dimensional complex vector space with nondegenerate bilinear form "g". The Clifford algebra is the algebra generated by "V" along with the anticommutation relation It is an abstract version of the algebra generated by the gamma or Pauli matrices. If "V" = "n", with the standard form we denote the Clifford algebra by Cℓ"n"().