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I am trying to solve an exercise asking to determine if $\mathfrak{so}(1,3)$ is simple or semisimple as real Lie algebra but I am having troubles.

My idea is to prove $\mathfrak{so}(1,3)$ is simple by using $\mathfrak{so}(1,3)\simeq \mathfrak{sl}(2,\mathbb{C})$ and $\mathfrak{sl}(2,\mathbb{C})=\mathfrak{su}(2)\oplus i\mathfrak{su}(2)$. Then according to my intuition ideals of $\mathfrak{sl}(2,\mathbb{C})$ should be of the form $\mathfrak{I}\oplus i\mathfrak{J}$, where $\mathfrak{I},\mathfrak{J}$ are ideals of $\mathfrak{su}(2)$, but I can't prove this rigorously. If this holds then I am able to conclude using that $\mathfrak{su}(2)$ is simple.

Thank you for any help

glS
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Eogan_96
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Yes, $\mathfrak{so}(1,3)$ is isomorphic as a $6$-dimensional real Lie algebra to $(\mathfrak{sl}_2(\mathbb{C}))_{\mathbb{R}}$ or $\mathfrak{su}(2)_{\mathbb{C}}$, see here, and here, and the links in these questions. This should help you with the exercise (note that if $L$ is the Lie algebra of a compact real Lie group then $L$ is simple if and only if $L_{\mathbb{C}}$ is simple).

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Dietrich Burde
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  • Thank you. Where can I find a proof of the fact "if $L$ is the Lie algebra of a compact real Lie group then $L$ is simple if and only if $L_{\mathbb{C}}$ is simple"?I looked in Hall-Lie Groups, Lie Algebras, and Representations but it is omitted. – Eogan_96 May 17 '16 at 14:33
  • It is for example in Kirillow's lecture notes. But you can show this directly, in the same way as for, say, $\mathfrak{sl}_2(\mathbb{R})$. – Dietrich Burde May 17 '16 at 16:31