Derivation in an algebra

A derivation is a linear map $D$ of the algebra into itself that satisfies the Leibniz rule:

where $a$ and $b$ are elements of the algebra.

Inner derivation

An inner derivation is a special kind of derivation that is defined on an associative algebra with a fixed unit element. An inner derivation is associated with an element of the algebra, and is defined as follows:
For any element $a$ in the algebra, the inner derivation associated with $a$, denoted by $\mathrm{ad}(a)$, is defined as the linear map on the algebra given by

where $[a,b]$ denotes the commutator of $a$ and $b$. It can be also denoted by $[-,a]$.