Examples of derivation of Lie algebras

The easiest example is perhaps to consider all derivations of the Heisenberg Lie Algebra $\mathfrak{h}_3(K)$, i.e., all linear maps $D\colon \mathfrak{h}_3(K) \rightarrow \mathfrak{h}_3(K)$ satisfying $D([x,y])=[D(x),y]+[x,D(y)]$ for all $x,y$. Here the brackets are given by $[e_1,e_2]=-[e_2,e_1]=e_3$, where $(e_1,e_2,e_3)$ denotes a basis. The inner derivations are of the form $ad (x)$, and are linear combinations of $$ ad (e_1)=\begin{pmatrix} 0 & 0 & 0\cr 0 & 0 & 0\cr 0 & 1 & 0\cr \end{pmatrix},\; ad (e_2)=\begin{pmatrix} 0 & 0 & 0\cr 0 & 0 & 0\cr -1 & 0 & 0\cr \end{pmatrix},\, ad(e_3)=0. $$ However, the Heisenberg Lie Algebra has many other derivations (outer derivations). In fact, all linear maps of the form $$ D=\begin{pmatrix} d_1 & d_4 & 0\cr d_2 & d_5 & 0\cr d_3 & d_6 & d_1+d_5\cr \end{pmatrix} $$ are derivations of the Heisenberg Lie algebra.

I'll venture an example, at the risk of being too trivial.

The zero Lie bracket makes the real polynomials $\Bbb R[x]$ into a Lie algebra, and any inner derivation with respect to this bracket would have to be uniformly zero.

But ordinary differentiation is a nonzero derivation of real polynomials, so this would furnish an example.

An instructive example is to pick a field $k$ and the commutative algebra $A=k[X]/(X^n)$. Since $A$ is commutative, its only inner derivation is the zero map, so any example of derivation you can come up with will be outer.

Write down explicitly the conditions for a linear map $A\to A$ to be a derivation, and solve them :-)