The complex linear representations of the complexification of formula_44 obtained via isomorphisms in , stand in one-to-one correspondence with the real linear representations of formula_27 The set of all "real linear" irreducible representations of formula_22 are thus indexed by a pair . The complex linear ones, corresponding precisely to the complexification of the real linear formula_18 representations, are of the form , while the conjugate linear ones are the . All others are real linear only. The linearity properties follow from the canonical injection, the far right in , of formula_22 into its complexification. Representations on the form or are given by "real" matrices (the latter are not irreducible). Explicitly, the real linear -representations of formula_22 are