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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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Computing the index of a Lie algebra: what is known beyond the reductive case?
The following paper of M. Rais has a formula for index of semi-direct products that you mentioned.
M. Rais "L’indice des produits semi-directs $g\ltimes E$.
C. R. Acad. Sci. Paris S´er. A-B 287 (19 …