# Reductive Levi decomposition of a parabolic subalgebra

Let $$\mathbb{K}$$ be a field (not assumed to be algebraically closed, but we can assume characteristic 0 if necessary), and let $$\mathfrak{g}$$ be a semisimple Lie algebra.

A Lie subalgebra $$\mathfrak{p} \leq \mathfrak{g}$$ is said to be parabolic if $$\mathfrak{p}^\perp = \operatorname{nil} \mathfrak{p},$$ where $$\mathfrak{p}^\perp$$ is the Killing perp, and $$\operatorname{nil} \mathfrak{p}$$ is the nilradical of $$\mathfrak{p}$$ (the unique maximal nilpotent ideal in $$\mathfrak{p}$$).

The nilradical gives us a short exact sequence $$0 \to \operatorname{nil} \mathfrak{p} \to \mathfrak{p} \to \mathfrak{p}^{\text{red}} \to 0$$ Where $$\mathfrak{p}^{\text{red}} = \mathfrak{p} / \operatorname{nil} \mathfrak{p}$$ is reductive. I have seen it mentioned in various places that this short exact admits a splitting on the right in the special case that $$\mathfrak{p}$$ is parabolic. Just naively, it seems like this should have to do with using the Killing form, since $$\mathfrak{p}^\perp = \operatorname{nil} \mathfrak{p}$$. But the Killing form is in general degenerate on $$\mathfrak{p}$$, so I'm not quite sure how to go about this.

Is there a way to construct this splitting (in a manner as choice-free as possible) without extending to an algebraically closed field $$\overline{\mathbb{K}}$$?

• A Cartan $\mathfrak h$ of $\mathfrak g$ which is contained in $\mathfrak p$ maps isomorphically to a Cartan of $\mathfrak p^{\mathrm{red}}$. If we have a Cartan which splits in $\mathfrak g$ then the quotient map to $\mathfrak p^{\mathrm{red}}$ is an isomorphism on root spaces $\mathfrak p_\alpha$ such that $\mathfrak p_{\alpha},\mathfrak p_{-\alpha}\neq 0$, and is the $0$ map otherwise. So we get a splitting $\mathfrak p^{\mathrm{red}}\to \mathfrak p$ by reversing the isos on each $\mathfrak p_{\alpha}$ and $\mathfrak h$. I can't speak to the non-split case, though. Nov 24 at 6:29

There are many such splittings. Any complementary (aka opposite) parabolic subalgebra (i.e. $$\mathfrak{q}$$ such that $$\mathfrak{p} \oplus \mathfrak{q}^\perp = \mathfrak{g}$$ and so on) provides a unique splitting $$\mathfrak{p} = \mathfrak{p} \cap \mathfrak{q} \oplus \mathfrak{p}^\perp$$. The space of complementary parabolic subalgebras is a $$\exp \mathfrak{p}^\perp$$-torsor or in other words an affine space modelled on $$\mathfrak{p}^\perp$$.
We can also equate these to a particular adjoint orbit in $$\mathfrak{p}$$. There is a particular element $$\xi^\mathfrak{p}_\mathfrak{q}$$ of $$\mathfrak{p}$$ called the grading element (aka canonical element) for which $$\mathfrak{q}^\perp \oplus \mathfrak{p} \cap \mathfrak{q} \oplus \mathfrak{p}^\perp$$ are the $$-1,0,1$$ eigenspaces for $$\mathrm{ad}\ \xi^\mathfrak{p}_\mathfrak{q}$$. This uniquely identifies the complementary subalgebra and the splitting.
Edit: an additional point is that the span of $$\xi^\mathfrak{p}_\mathfrak{q}$$ is exactly the 1-dimensional centre of $$\mathfrak{p} \cap \mathfrak{q}$$ and its orthocomplement is semisimple.