= Split semisimple Lie algebra
{title2=$\mathfrak g=\mathfrak h\oplus\bigoplus_{\alpha}\mathfrak g_\alpha\quad\text{over the ground field}$}
A <semisimple Lie algebra> over a field of <characteristic zero> is split if it has a <Cartan subalgebra> whose adjoint action is simultaneously diagonalizable over that field. Its <root-space decomposition> then needs no extension of scalars. The <sl2R Lie algebra> is split, with the real diagonal <Cartan subalgebra>. The real <special unitary Lie algebra> $\mathfrak{su}(2)$ is not: its adjoint operators are skew-adjoint for the positive definite <inner product> obtained by negating its <Killing form>. Thus a split adjoint operator would have to be zero.
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