generators of SU(2) can be written as Pauli matrices. In 4 Euclidean dimensions, the corresponding isomorphism is . There are two inequivalent quaternionic 2-component Weyl spinors and each of them transforms under one of the SU(2) factors only. In 5 Euclidean dimensions, the relevant isomorphism is that implies that the single spinor representation is 4-dimensional and quaternionic. In 6 Euclidean dimensions, the isomorphism guarantees that there are two 4-dimensional complex Weyl representations that are complex conjugates of one another. In 7 Euclidean dimensions, the single spinor representation is 8-dimensional and real; no isomorphisms to a Lie algebra from another series (A or C) exist from this dimension on. In 8 Euclidean dimensions, there are two Weyl–Majorana real 8-dimensional representations that are related to the 8-dimensional real vector representation by a special property of Spin(8) called triality. In dimensions, the number of distinct irreducible spinor representations and their reality (whether they are real, pseudoreal, or complex) mimics the structure in d dimensions, but their dimensions are 16 times larger; this allows one to understand all remaining cases. See Bott periodicity. In spacetimes with p spatial and q time-like directions, the dimensions viewed as dimensions over the complex numbers coincide with the case of the -dimensional Euclidean space, but the reality projections mimic the structure in Euclidean dimensions. For example, in dimensions there are two non-equivalent Weyl complex (like in 2 dimensions) 2-component (like in 4 dimensions) spinors, which follows from the isomorphism . See also Anyon Dirac equation in the algebra of physical space Eigenspinor Einstein–Cartan theory Projective representation Pure spinor Spin-1/2 Spinor bundle Supercharge Twistor theory Notes References Further reading Rotation in three dimensions Quantum mechanics Quantum field theory