  We consider the approach of replacing trees by multi-indices as an index set
of the abstract model space $\mathsf{T}$ introduced by Otto, Sauer, Smith and
Weber 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
$\Delta\colon\mathsf{T}\rightarrow \mathsf{T}^+\otimes\mathsf{T}$. In fact,
this approach, where the dual $\mathsf{T}^*$ of the abstract model space
$\mathsf{T}$ naturally embeds into a formal power series algebra, 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 on this power series algebra. These derivations in turn are the
infinitesimal generators of two actions on the space of pairs (nonlinearities,
functions of space-time mod constants). These actions are shift of space-time
and tilt by space-time polynomials. 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 are determined
by an underlying pre-Lie algebra structure of the derived algebra of
$\mathsf{L}$. Strong finiteness properties, which are enforced by gradedness
and the restrictive definition of $\mathsf{T}$, allow for this purely algebraic
construction of $\mathsf{G}$. We also argue that there exist pre-Lie algebra
and Hopf algebra morphisms between our structure and the tree-based one in the
cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the
generalized parabolic Anderson model.
