What is the definition of derivation of tensor algebra?

Question

Derivation $d$ of a Lie algebra $L$ is defined as $d([x,y])=[d(x),y]+[x,d(y)]$ for all $x,y \in L$. Let consider $T(L)$ be the tensor algebra and $S(L)$ be the symmetric algebra of $L$. How a derivation map on $T(L)$ is defined?

Answer 1

Let $A$ be a $K$-algebra, not necessarily associative, with $K$-bilinear product $(x,y)\mapsto x\cdot y$. A derivation of $A$ is a $K$-linear map $D\colon A\rightarrow A$ satisfying the Leibniz rule $$ D(x\cdot y)=D(x)\cdot y+x\cdot D(y) $$ for all $x,y\in A$. The tensor algebra $A=T(V)$ is an associative algebra, so the definition also applies here. The space of all derivations of $A$ is denoted by ${\rm Der}(A)$. It forms a Lie subalgebra of the linear Lie algebra $\mathfrak{gl}(A)$.