Remark 2

If $k$ is algebraically closed, every semi-simple Lie algebra $\mathfrak{g}$ is splittable, and every Cartan subalgebra of $\mathfrak{g}$ is splitting. When $k$ is not algebraically closed, there exist non-splittable semi-simple Lie algebras (Exerc. $2a$)$)$; moreover, if $\mathfrak{g}$ is splittable, there may exist Cartan subalgebras of $\mathfrak{g}$ that are not splitting (Exerc. $2b$)$)$.