"Semi-simplicity" of Lie algebra elements.

Question

Why are diagonalizable elements of Lie algebra called "semi-simple"?
Is there a notion of "simple" elements? Is it related to "semi-simplicity" of the Lie algebra?

Answer 1

As far as I understand, this terminology is used as an analogy of what happens with matrices, and is related to the notion of semi-simple module mentioned in the comments.

Consider an $n\times n$ matrix $A$ over the complex numbers. Then $A$ defines a structure of $\mathbb{C}[X]$-module over $\mathbb{C}^n$, by letting $X$ act by $A$. Now, this module is semi-simple (that is, it is a direct sum of simple (or irreducible) modules) if and only if the matrix $A$ is diagonalizable. Indeed, $A$ is conjugate to a Jordan normal form, and each Jordan block defines an indecomposable direct summand of the module. It is not hard to see that a Jordan block defines a simple module if and only if it is one-dimensional. Hence the module is semi-simple if and only if all Jordan blocks have size 1, which means that $A$ is diagonalizable.

For abstract Lie algebras, an element is then semi-simple if its adjoint action is given by a diagonalizable matrix.