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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.

bonif
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    Every complex simple Lie algebra viewed as a real Lie algebra is simple but not absolutely simple of which $\mathfrak{so}(3,1) \cong \mathfrak{sl}(2,\mathbb{C})$ is an example. – Callum Sep 27 '23 at 21:08
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    In fact, if I am reading Torsten's answer here correctly, then these are the only examples. – Callum Sep 27 '23 at 21:16
  • @Callum I can see from the answer linked there are no other compact example, I'm not so sure about non-compact. – bonif Sep 27 '23 at 21:33
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    Well a complex Lie algebra certainly isn't compact viewed as a real Lie algebra. Indeed it won't be split either, although it will be quasi-split. The linked answer claims that there are no other examples besides these complex Lie algebras viewed as real Lie algebras. – Callum Sep 27 '23 at 22:51
  • Got it, thanks. – bonif Sep 27 '23 at 22:53

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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.

Torsten Schoeneberg
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