I've been stuck on this problem for quite a while, and I can't seem to find any similar questions, though if my terminology is wrong and that's why I failed to find one then my apologies.
Let $A, B \in \mathfrak{se}(3)$ be members of the (vector-representation) Lie algebra corresponding to the Special Euclidean Group of 3 dimensions $SE(3)$. We will use orientation-first representation such that a member representing position would be $X \in \mathfrak{se}(3) := \begin{pmatrix} \theta \\ x \end{pmatrix} \in \mathbb{R}^6$.
Let $\mathbf{C} \in \mathbb{R}^{6x6}$ be an arbitrary 6 by 6 square matrix
Define $ad_X := \begin{pmatrix} [\theta] & 0 \newline [x] & [\theta] \end{pmatrix}$ be the adjoint of the vector space. The coadjoint is likewise defined $ad_X^* \equiv ad_X^T$.
Given the expression $ad_A^* \mathbf{C} B$, is it possible to rearrange the matrix-matrix-vector expression to an alternative matrix-vector expression of the format $\mathbf{D}A$? That is, can the expression be rearranged such that a set of known, fixed matrices constructed from known, fixed parameters $B, \mathbf{C}$ can be applied to a known but time-varying parameter vector $A$ for use in e.g. Quadratic Programming based optimisation techniques?
Similar to how the Kronnecker product + vectorisation function can be used to transform a matrix-matrix-matrix equation into a matrix-vector equation as per: $vec(\mathbf{HJK}) \equiv (\mathbf{K}^T \otimes \mathbf{H}) vec(\mathbf{J})$, which is useful if $J$ is constructed from time-varying parameters.