I am currently studying a memoir written by Anderson and Thompson on the inverse problem of the calculus of variations [1]. In particular, I am working through Example 6.4, where they prove that if the Jacobi endomorphism $\Phi$ of a second-order system $\ddot{x}^i=F^i(t,u^j,\dot{u}^j)$ (with coordinates on $E:=\mathbb{R}\times TM$ where $M$ is an $n$-dimensional manifold) is a multiple of the identity tensor, then a Lagrangian exists for that system. At one point in this computation (eq. (6.26)), they claim that a certain $2$-form can be written as $$\pi_{ijh}\wedge \Theta^h+\rho_{ijh}\wedge \Psi^h\, ,$$ where $\pi_{ijh}$ and $\rho_{ijh}$ are one-forms symmetric in the indices $ijh$, and $\Theta^h:=du^h-\dot{u}^h~dt$ and $\Psi^h:=d\dot{u}^h-F^h~dt-\frac{1}{2}\frac{\partial F^h}{\partial \dot{u}^j}\Theta^j$ are linearly independent $1$-forms. I have managed to show that all terms in equation (6.27) can be written in that form, except for the following two terms (let's denote it by $\tau_{ij}$): $$\tau_{ij}=-\frac{1}{6}\left[r_{lj}\Phi_{hik}^l+r_{il}\Phi_{hjk}^l\right] \Theta^h\wedge \Theta^k+\frac{1}{2}\left[r_{lj} G_{ihk}^l+r_{il} G_{jhk}^l\right] \Psi^k \wedge \Theta^h\, . \tag{1}$$
In all these expressions we use the Einstein summation convention and all indices go from $\{1,\dots,n\}$. Here's what we know:
We assume that $\Phi$ is a multiple of the identity, i.e. $\Phi_j^i:=\frac{d}{dt}\left(\frac{\partial F^i}{\partial \dot{u}^j}\right)-2\frac{\partial F^i}{\partial u^j}-\frac{1}{2}\frac{\partial F^i}{\partial \dot{u}^k}\frac{\partial F^k}{\partial \dot{u}^j}=\mu \delta_j^i$ where $\mu$ is an arbitrary smooth function on $E$. This implies that $\Phi_{hik}^l:=\frac{\partial^2 \Phi_h^l}{\partial \dot{u}^i \partial \dot{u}^k}=\frac{\partial^2 \mu}{\partial \dot{u}^i \partial \dot{u}^k}\delta_h^l$.
$G_{jhk}^l:=\frac{\partial^3 F^l}{\partial \dot{u}^j \partial \dot{u}^h \partial \dot{u}^k}$. In particular, it is symmetric in $jhk$.
$r_{ij}$ is symmetric, i.e. $r_{ij}=r_{ji}$ for all $i,j\in \{1,\dots,n\}$.
Here's an attempt at dealing with the first term in $(1)$. We see that $$\begin{align} -\frac{1}{6}\left[r_{lj}\Phi_{hik}^l+r_{il}\Phi_{hjk}^l\right] \Theta^h\wedge \Theta^k&=-\frac{1}{6}r_{lj} \frac{\partial^2 \mu}{\partial \dot{u}^i \partial \dot{u}^k}\delta_h^l \Theta^h\wedge \Theta^k-\frac{1}{6} r_{il} \frac{\partial^2 \mu}{\partial \dot{u}^j \partial \dot{u}^k} \delta_h^l \Theta^h\wedge \Theta^k\\&=-\frac{1}{6}r_{hj} \frac{\partial^2 \mu}{\partial \dot{u}^i \partial \dot{u}^k} \Theta^h\wedge \Theta^k-\frac{1}{6} r_{ih} \frac{\partial^2 \mu}{\partial \dot{u}^j \partial \dot{u}^k} \Theta^h\wedge \Theta^k\\&=-\frac{1}{6}r_{hj} \frac{\partial^2 \mu}{\partial \dot{u}^i \partial \dot{u}^k} \Theta^h\wedge \Theta^k+\frac{1}{6} r_{ik} \frac{\partial^2 \mu}{\partial \dot{u}^j \partial \dot{u}^h} \Theta^h\wedge \Theta^k\\&=\frac{1}{6}\left[r_{hj} \frac{\partial^2 \mu}{\partial \dot{u}^i \partial \dot{u}^k}-r_{ik} \frac{\partial^2 \mu}{\partial \dot{u}^j \partial \dot{u}^h}\right]\Theta^k\wedge \Theta^h \end{align}$$ Suppose we denote $\hat{\pi}_{ijh}:= \frac{1}{6}\left[r_{hj} \frac{\partial^2 \mu}{\partial \dot{u}^i \partial \dot{u}^k}-r_{ik} \frac{\partial^2 \mu}{\partial \dot{u}^j \partial \dot{u}^h}\right]\Theta^k$. Then clearly we have symmetry in $jh$, but e.g. symmetry does not hold in $ij$. As such, $\hat{\pi}_{ijh}$ is not totally symmetric.
In summary, the objective is to determine totally symmetric $1$-forms $\pi_{ijh}$ and $\rho_{ijh}$ such that $\tau_{ij}=\pi_{ijh}\wedge \Theta^h+\rho_{ijh}\wedge \Psi^h$ where $\pi_{ijh}$ and $\rho_{ijh}$ are totally symmetric.
[1] Anderson, I., & Thompson, G. (1992). The inverse problem of the calculus of variations for ordinary differential equations. American Mathematical Society.