Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 302 / 2

A Cartan subalgebra $\mathfrak{h}$ of a finite-dimensional complex Semisimple Lie algebra is a maximal abelian subalgebra consisting of semisimple elements. The root-space decomposition isand the nonzero functionals $\alpha$ are the roots. A Cartan-Weyl basis consists of a basis $H_i$ of $\mathfrak{h}$ and root vectors $E_{\alpha }\in {\mathfrak{g}}_{\alpha }$. Its brackets have the formand $[E_{\alpha },E_{\beta }]=N_{\alpha \beta }{E}_{\alpha +\beta }$ when $\alpha +\beta$ is a root, and zero when $\alpha +\beta$ is neither a root nor zero.

For the complexified so4 Lie algebra, take $H^1=T^{(12)}$ and $H^2=T^{(34)}$. WriteDirect use of the stated commutation relations givesThe simultaneous eigenvectors, hence the step generators, may be chosen asThus the roots relative to $(H^1,H^2)$ are $(\pm i,\pm i)$. Replacing $H^a$ by $-i{H}^a$ gives the usual real coordinates $(\pm 1,\pm 1)$. The only nonzero brackets between step generators, apart from those obtained by antisymmetry, are

An isomorphism of Lie algebras is a bijective linear map preserving the Lie bracket. DefineThenThe two spans are commuting copies of the complexified ${\mathfrak{su}}_2$, and together contain all six basis elements of ${\mathfrak{so}}_4$. Chiral decomposition of the complexified so4 Lie algebra therefore givesUnder this isomorphism the Adjoint representation is the direct sum of the adjoint representations of the two factors: