 Split semisimple Lie algebra

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.