Are there any other examples of simple not absolutely simple lie algebra besides $so(3,1)$?

Question

I just learned the distinction between simple and absolutely simple Lie algebras and was wondering if there are other well known examples other than the Lorentz algebra.

Answer 1

Just to get this off the "unanswered" list:

If $\mathfrak h$ is a simple real Lie algebra that is not absolutely simple (i.e. the complexification $\mathbb C \otimes_{\mathbb R} \mathfrak h$ is not simple), then $\mathfrak h$ is the scalar restriction of a complex simple Lie algebra, i.e. there exists a complex simple Lie algebra $\mathfrak g$ such that $\mathfrak h \simeq Res_{\mathbb C|\mathbb R} \mathfrak g$ as real Lie algebras -- where the admittedly cumbersome notation $Res_{\mathbb C|\mathbb R}$ means scalar restriction i.e. forgetting the complex structure i.e. what some people call "realification".

(For a proof sketch, see my answer to On a simple real Lie algebra whose complexification is not simple, for a more detailed proof in the case of far more general field extensions see my answer to Do endomorphisms of the adjoint representation of a Lie algebra commute?. To my knowledge, at least the general case is due to N. Jacobson.)

And conversely, each complex simple Lie algebra $\mathfrak g$, when considered as a real Lie algebra $Res_{\mathbb C|\mathbb R} \mathfrak g$, is simple but not absolutely simple.

So those complex-simple-viewed-as-real Lie algebras are all examples. The smallest one, the six-dimensional real $Res_{\mathbb C|\mathbb R} \mathfrak{sl}_2(\mathbb C)$, stands out in the sense that it happens to be isomorphic to another "classical" Lie algebra, namely $\mathfrak{so}(1,3)$. This exceptional isomorphism, or rather, the fact that there is an exceptional second name for $Res_{\mathbb C|\mathbb R} \mathfrak{sl}_2(\mathbb C)$, boils down to the fact that the root system $A_1 \times A_1$ can also be called $D_2$. While for any other simple root system $R$, its product with itself $R \times R$ does not have any other name elsewhere in the list of root systems.