Understanding derivations on a Lie algebra I am very new to the subject of Lie algebras so excuse me if this is a rather stupid question.
I have some trouble understanding the Leibniz rule of a derivation $\delta$ on some Lie algebra $L$. 
So a derivation $\delta$ is a linear map $\delta:L\rightarrow L$
that fullfills the Leibniz rule, i.e. $\delta(ab)=\delta(a)b+a\delta(b)$.
And here is my problem, how is the product $ab$ defined on a general vector space? 
For $L=C(\mathbb{F})$ I know that $ab$ is just the product of two functions. But what do I do if $L=\mathbb{F}^n$? How to I define a product there?
 A: You are new to Lie algebras, and not to arbitrary algebras with product $ab$.
So here $ab$ is the Lie bracket $[a,b]$ and the Leibniz rules then reads as
$$
D([a,b])=[D(a),b]+[a,D(b)]
$$
for all $a,b\in L$. For the definition of a  Lie algebra see here.
For a $K$-algebra $A$ the product on the vector space $A$ is given by the algebra, i.e., by the bilinear map $A\times A\rightarrow A$, $(a, b)\mapsto ab$.
A: On a Lie algebra $L$, the "multiplication" is the bracket operation $[\cdot,\cdot]$ that $L$ is equipped with, which must be bilinear, alternating, and satisfy the Jacobi identity (by definition). Usually, $L$ is a vector space of square matrices (or linear maps from a vector space to itself), equipped with the "commutator bracket" $[A,B]=AB-BA$.
A: The product $ab$ is not defined on an arbitrary vector space. If you have a bilinear map\begin{array}{ccc}L\times L&\longrightarrow&L\\(a,b)&\mapsto&ab\end{array}then a linear map $\delta$ from $L$ into itself is a derivation with respect to that bilinear map if$$(\forall a,b\in L):\delta(ab)=\delta(a)b+a\delta(b).$$
