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Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
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Motivating the Casimir element
I think that there are two things to be motivated: one is the Casimir, and the other is the proof of semi-simplicity.
First, for the Casimir, it might help to note that there is a ``formula-free" con …