Given a manifold $ M $ and a $ \mathrm TM $-valued $ (k + 1) $-form $ K $ on $ M $ one can define a $ k $-derivation $ i_K $ of $ \Omega(M) $ by some messy formula. The $ i_K $ should reduce to the interior product when $ K\in \Omega^0(M;\mathrm TM) $, i.e. when $ K $ is a vector field on $ M $. I'm trying to tackle this construction. The reference I'm using is Chapter 1, Section 8 of Natural Operations in Differential Geometry.
Let's start by fixing some terminology. Given $ k\in \mathbb Z $, a $ k $-derivation of a graded commutative $ \mathbb R $-algebra (GCA) $ A = \bigoplus_k A_k $ is a $ \mathbb R $-linear map $ D\colon A\to A $ such that $ Da\in A_{k + l} $ for every $ a\in A_l $, and such that $ D(ab) = D(a)b + (-1)^{kl}aD(b) $ for every $ a\in A_l $ and every other $ b\in A $. A $ k $-derivation is called algebraic if $ Da = 0 $ for all $ a\in A_0 $. The set of all $ k $-derivations of $ A $ is denoted by $ \mathrm{Der}_k(A) $, and we put $ \mathrm{Der}(A) = \bigoplus_k\mathrm{Der}_k(A) $.
Now, let $ M $ be a manifold. Let $ K\in \Omega^{k+1}(M;\mathrm TM) $ be a $ \mathrm TM $-valued differential $ (k + 1) $-form on $ M $. The authors claim that $ K $ induces a $ k $-derivation $ i_K $ of the GCA $ \Omega(M) $ of differential forms on $ M $ in the following way. First define maps $ i_k^{(l)}\colon \Omega^l(M)\to \Omega(M) $ by $$ i_K^{(l)}\omega(X_1,\dots,X_{k + l}) = \frac{1}{(k + 1)!(l - 1)!}\sum_{\sigma\in S_{k + l}}\operatorname{sign}\sigma\, \omega\bigl(K(X_{\sigma(1)},\dots,X_{\sigma(k + 1)}),X_{\sigma(k + 2)},\dots,X_{\sigma(k + l)}\bigr)\text{,} $$ where $ \omega\in \Omega^l(M) $ and the $ X_i $s are vector fields on $ M $, and then glue (?) them together using the universal property of the direct sum $ \bigoplus_k\Omega^k(M) $.
I don't understand their proof that this actually defines a derivation. Apparently, one should be able to derive this messy formula for $ i_k $ from the fact that $ \bigwedge \mathrm T_p^*M = \bigoplus_k \bigwedge^k\mathrm T_p^*M $ is the free graded commutative algebra over the vector space $ \mathrm T_p^*M $ (in the sense that the functor mapping $ V\mapsto \bigwedge V $ from vector spaces to DGAs is the left adjoint to the functor $ A = \bigoplus_kA_k\mapsto A_1 $ mapping a GCA $ A $ to its vector space of degree-$ 1 $ elements), just by inductively "applying $ K $ to an exterior product of $ 1 $-forms" (semi-quote), but it's not clear to me in principle how to do that.
