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I'm studying the article of Kac "Lie Superalgebras" (1974) and in several times he use the fact that if $L$ is a semisimple Lie Algebra, $V$ its faithful, irreducible and finite-dimensional module with $\lambda$ highest weight and $\mu$ lowest weight. Then if $2\lambda$ or $\lambda - \mu$ are a root of $L$ then $L$ must be simple. I have no clue why this is true, can someone help me?

Lorenzo Vittori
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    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community May 12 '22 at 19:08

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