As an automorphism of "G", "σ" fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra formula_6 of "G", also denoted by "σ", whose square is the identity. It follows that the eigenvalues of "σ" are ±1. The +1 eigenspace is the Lie algebra formula_7 of "H" (since this is the Lie algebra of "G""σ"), and the −1 eigenspace will be denoted formula_8. Since "σ" is an automorphism of formula_6, this gives a direct sum decomposition