Consider a Lie algebra. The ladder operators (i.e. root vectors, or eigenvectors of the Cartan subalgebra with respect to the adjoint representation) form a handy basis of the algebra called a Cartan-Weyl basis (not unique).
But in the following example of su(2) they are not antihermitian, while su(2) requires that.
Consider su(2). As a vector space it can be the real vector space of traceless antihermitian matrices. The Cartan subalgebra is spanned by
$H = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$.
The ladder operators are
$\begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix}$ and $\begin{pmatrix} 0 & 0 \\ -2 & 0 \end{pmatrix}$.
But they are not antihermitian. How is it that they can be used as a basis for the su(2) algebra?