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For example, the Lie algebras of $SO(4)$ and $SO(3,1)$ are the same, called $D_2$ and isomorphic to $A_1 \times A_1$ ($= \mathfrak{su}(2)\times \mathfrak{su}(2)$).

Thus $SO(4)$ is a ([...] compact?) form of $D_2$ and $SO(3,1)$ is a ([...] non-compact?) form of $D_2$

In addition, for almost every Lie group the Wikipedia article talks about the various forms of the Lie algebra (e.g. here).

What does it really mean that there are several (non-isomorphic) groups which are "forms of the same Lie algebra"?

jak
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  • See compact real form of a complex semisimple Lie algebra. – Dietrich Burde Dec 19 '16 at 15:09
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    Correctly speaking, a real form of a complex Lie algebra should be a real Lie algebra with the former as complexification). Referring to the groups as "forms" is just informal language. – YCor Dec 20 '16 at 04:38
  • @YCor Thanks,, this helps me! – jak Dec 20 '16 at 07:02
  • It is not true that the Lie algebras of $SO(4)$ (called $\mathfrak{so}4$) and $SO(3,1)$ (called $\mathfrak{so}{3,1}$) are the same. The first one is isomorphic to $\mathfrak{su}_2 \oplus \mathfrak{su}_2$, the second is not. Their complexifications are isomorphic, namely both isomorphic to $\mathfrak{sl}_2(\mathbb C) \oplus \mathfrak{sl}_2(\mathbb C)$. Which exactly means that those two (real) Lie algebras are real forms of this complex Lie algebra. Compare https://math.stackexchange.com/a/3258221/96384 and https://math.stackexchange.com/q/639749/96384. – Torsten Schoeneberg Jul 23 '19 at 03:57

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