Abstract
The Lie algebras opened a vast field of applications in physics, differential equations, differential geometry, linear algebra and abstract algebra. Below is a demonstration of the Engel’s theorem. This theorem is one of the foundations of the theory of Lie algebras and, in itself corresponds to a general property of linear algebra. In particular Engel’s theorem states that a set of linear nilpotent transformations in a finite dimensional vector space closed for the Lie bracket has a common eigenvector associated to the eigenvalue zero.
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