  In this work, we approach the Laplacian coflow of a coclosed $G_2$-structure
$\varphi$ using the formulae for the irreducible $G_2$-decomposition of the
Hodge Laplacian and the Lie derivative of the Hodge dual $4$-form of $\varphi$.
In terms of this decomposition, we characterise the conditions for a vector
field as an infinitesimal symmetry of a coclosed $G_2$-structure, as well as
the soliton condition for the Laplacian coflow. More specifically, we provide
an easier proof for the absence of compact shrinking solitons of the Laplacian
coflow. Moreover, we revisit the Laplacian coflow of coclosed $G_2$-structures
on almost Abelian Lie groups addressed by Fino-Bagaglini (2018). However, our
approach is based on the bracket flow point of view. Notably, by showing that
the norm of the Lie bracket is strictly decreasing, we prove that we have
long-time existence for any coclosed Laplacian coflow solution.
