TITLE: Characterization of the Lie derivative
QUESTION [14 upvotes]: The exterior differential of differential forms on a manifold can be characterized as the unique super-derivation of degree 1 on the exterior algebra of forms such that $<df,X>=X(f)$
for $f$ a $C^{\infty}$ function, $X$ a vector field. So we really only need to know how to compute $df$, and everything else follows formally.
Is there a similar characterization for the Lie derivative acting on differential forms? 
My guess: Extend $L_{X}$ from $L_{X}f=X(f)$ to an action on all forms so that it commutes with $d$ (Cartan's magic formula?) and perhaps $L_{[X,Y]}=[L_{X},L_{Y}]$. In other words, the Lie derivative should be a homomorphism of Lie algebras from vector fields to degree zero derivations of the de Rham algebra. 
If this is correct, can anyone give references for this point of view?

REPLY [2 votes]: Let me add my two cents to the great answers of Dick Palais and Tom Goodwillie. Let $A^\bullet(M)=\bigoplus_{k=0}^\infty A^k(M)$ be the graded algebra of differential forms on your manifold $M$. Then, for a vector field $X$, $\iota_X$ is the unique derivation of degree $(-1)$, satisfying $\iota_X(\alpha)=\alpha(X)$ for all $\alpha\in A^1(M)$, and $d$ is the unique derivation of degree $(+1)$, satisfying $df(X)=X(f)$ for all $f\in A^0(M)$.
If $D_1$ and $D_2$ are graded derivations of  degrees $p$ and $q$ respectively, their graded commutator
$$
D_1D_2 - (-1)^{pq}D_2D_1
$$
 is a derivation of degree $p+q$. In our case, this is
$$
L_X = d\iota_X + \iota_X d,
$$
a degree zero derivation.