What is the relationship between semisimple lie algebras and semisimple elements?

A Lie algebra $\mathfrak{g}$ is said to be semisimple if its radical is zero. An element $x \in \mathfrak{g}$ is said to be semisimple if $\text{ad} x$ is diagonalizable.

A complex semisimple Lie algebra must contain non-zero semisimple elements. But is there any deeper connection underlying the common names?

(For instance, in the theory of algebraic groups, a separable element of a matrix group (one with distinct eigenvalues) is one that generates a separable algebra.)

Solution 1:

In each name, the word "semisimple" means "a direct sum of simple objects" in the appropriate sense. It can be shown that a complex Lie algebra is semisimple (has radical zero) if and only if it is a direct sum of simple Lie algebras. On the other hand, saying that $\operatorname{ad} x$ is diagonalizable is the same as saying that $\mathfrak{g}$ decomposes into a direct sum of eigenspaces for $\operatorname{ad} x$, on each of which $\operatorname{ad} x$ acts as a scalar multiple of the identity; in other words, $\operatorname{ad} x$ is a "direct sum of scalars".