Root system of a semi-simple Lie algebra

In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semisimple Lie algebras. Here the construction of a root system of a semisimple Lie algebra – and, conversely, the construction of a semisimple Lie algebra from a reduced crystallographic root system – are shown.

Let ${\displaystyle {\mathfrak {g}}}$ be a complex semisimple Lie algebra. Let further ${\displaystyle {\mathfrak {h}}}$ be a Cartan subalgebra of ${\displaystyle {\mathfrak {g}}}$. Then ${\displaystyle {\mathfrak {g}}}$ acts on ${\displaystyle {\mathfrak {g}}}$ via simultaneously diagonalizable linear maps in the adjoint representation. For λ in ${\displaystyle {\mathfrak {h}}^{*},}$ define the subspace ${\displaystyle {\mathfrak {g}}_{\lambda }\subset {\mathfrak {g}}}$ by

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