Suppose I have (simple, compact) Lie group with Lie derivatives $\partial_i$. These do not commute, but instead it is $[\partial_i,\partial_j]=\partial_i\partial_j-\partial_j\partial_i=-f_{ijk}\partial_k$, with (totally antisymmetric) structure constants $f$ (and einstein sum conventions of course).
Now I have a lot of expressions involving higher derivatives, so I am looking for simplification rules. I already found \begin{align} \partial_i\partial_k\partial_i &= (\partial^2+\frac{1}{2}C_A)\partial_k \\\ \partial_i\partial_j\partial_k\partial_i\partial_j &= (\partial^2+C_A)(\partial^2+\frac{1}{2}C_A)\partial_k \\\ \end{align} where $C_A$ is the casimir operator in the adjoint representation (i.e. just a constant number which fullfills $f_{ijk}f_{ijl}=C_A\delta_{kl}$), and the operator $\partial^2=\partial_i\partial_i$ commutes with all derivatives. Now I am looking for generalizations of these, in particular \begin{align} \partial_i\partial_{k_1}...\partial_{k_n}\partial_i &= ? \\\ \partial_i\partial_j\partial_{k_1}...\partial_{k_n}\partial_i\partial_j &= ? \\\ \end{align}
Is there a good formula for those, i.e. something that does not explicitly contain the structure constants, but only simple constants like $C_A$?
(even just $n=2$ would be very helpful). Would be very glad if someone could point me to some literature for this kind of computation.