  We consider the approach of replacing trees by (fewer) multi-indices as an
index set of the abstract model space $\mathsf{T}$ to tackle quasi-linear
singular SPDEs. We show that this approach is consistent with the postulates of
regularity structures when it comes to the structure group $\mathsf{G}$. In
particular, $\mathsf{G}\subset{\rm Aut}(\mathsf{T})$ arises from a Hopf algebra
$\mathsf{T}^+$ and a comodule $\Delta\colon\mathsf{T}\rightarrow
\mathsf{T}^+\otimes\mathsf{T}$. This approach allows to interpret
$\mathsf{G}^*\subset{\rm Aut}(\mathsf{T}^*)$ as a Lie group arising from a Lie
algebra $\mathsf{L} \subset{\rm End}(\mathsf{T}^*)$ consisting of derivations;
these derivations in turn are the infinitesimal generators arising from actions
on the space of pairs (nonlinearities, functions of space-time). The Hopf
algebra $\mathsf{T}^+$ arises from a coordinate representation of the universal
enveloping algebra ${\rm U}(\mathsf{L})$ of the Lie algebra $\mathsf{L}$. The
coordinates arise from an underlying pre-Lie algebra structure of $\mathsf{L}$.
Strong finiteness properties, which are enforced by gradedness and the
restrictive definition of $\mathsf{T}$, allow to define these structures in our
infinite-dimensional setting.
