What is an example of semisimple Lie algebra which is not a simple Lie algebra?
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4Take the direct sum of two simple Lie algebras. – Angina Seng Jul 20 '18 at 05:50
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Or the zero Lie algebra, although the Lord's example is certainly more enlightening and useful. – Torsten Schoeneberg Jul 20 '18 at 15:59
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As pointed out in the comments, you can take any two simple Lie algebras, like $\mathfrak{sl}_5$ and $\mathfrak{so}_8$, and consider their direct sum $\mathfrak{g} = \mathfrak{sl}_5 \oplus \mathfrak{so}_8$. Then $\mathfrak{g}$ is semisimple but not simple.
Mike Pierce
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