Jacobi identity and Leibniz rule - the same thing?

Question

Is there any formal connection between the Jacobi identity $$[[a,b],c] = [a,[b,c]] + [b,[c,a]]$$ and the Leibniz rule $$d(a \cdot b) \cdot c = a \cdot d(b) \cdot c + b \cdot c \cdot d(a) ~\text{?}$$

Answer 1

Yes, with $d=\operatorname{ad}(x)(y)$, the adjoint operator in a Lie algebra, defined by $\operatorname{ad}(x)(y)=[x,y]$. Then the Jacobi identity is equivalent to the statement that the operator $d$ satisfies the Leibniz rule, with $x.y=[x,y]$: $$ d([a,b])=[d(a),b]+[a,d(b)] $$ Applying $\operatorname{ad}(c)$ yields $[d([a,b]),c]=[[d(a),b],c]+[[a,d(b)],c]$.