I'm working on the article of Benkart and Osborn " Flexible Lie-admissible algebras", specially, i'm working on the lemma 3.1, this lemma represent the dimension of L-module homomorphisms of $L\otimes L$ into L which equals the number of times L appears in the decomposition of $L\otimes L$ into irreducible L-modules. In the course of the proof of this lemma, i ' m trying to show the method but i don't understand how they got the adjoint decomposition represented in the table I and where appear the adjoint representation in this decompoaition. Also, how can i use the weyl's formula in this calculation.
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