What are examples of toral non-abelian Lie subalgebras?

Question

Following the Wiki's definition, given a semisimple Lie algebra $\mathfrak g$, a toral subalgebra is a subalgebra whose elements have semisimple adjoint operators (I'm not quite sure whether one defines toral subalgebras for more general cases, the wiki isn't very clear).

It is mentioned there, as well as in this other question, that when $\mathfrak g$ is defined over an algebraically closed field, any toral subalgebra is abelian. A sketch of the argument as I understand it being:

• given toral $\mathfrak t\le \mathfrak g$, and $x\in\mathfrak t$, if $\operatorname{ad}(x)\neq 0$ then it being a diagonalisable operator in an algebraically closed field, there's $y\neq0$ such that $[x,y]=\lambda y$ for some $\lambda\neq0$.
• thus $\operatorname{ad}(y)^2x=0$. But a semisimple (ie diagonalisable) matrix $A$ such that $A^2x=0$ must also satisfy $Ax=0$, which is a contradiction. So $[x,y]=0$ for all $x,y\in\mathfrak t$.

What are some examples of toral subalgebras (in non-algebraically-closed fields) that are not abelian?

Answer 1

As discussed in this answer, an easy example is the real algebra $\mathfrak{su}(2)$, defined as the real span of $i\sigma_x$, $i\sigma_y$, $i\sigma_z$, with $\sigma_j$ Pauli matrices.

This is toral because the adjoints of all its elements are semisimple. A way to see it is to note that each $\operatorname{ad}(x)$ acts as a 3x3 skew-symmetric matrix, and is thus semisimple. At the same time, $\mathfrak{su}(2)$ is obviously not abelian, hence it fits the prescription.

It's also worth noting that the complexification $\mathfrak{su}(2)$, namely $\mathfrak{sl}(2,\mathbb{C})$, is not toral, as not all of its elements are semisimple. For example, writing the generators as $[H,E_+]=2E_+$, $[H,E_-]=-2E_-$, $[E_+,E_-]=H$, we can observe that $\operatorname{ad}(E_+)$ is not semisimple, as $\operatorname{ad}(E_+)^3=0$. Of course, this has to be the case, as if $\mathfrak{sl}(2,\mathbb{C})$ were toral, it would have to be abelian by the result stated above.