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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 theHodge Laplacian and the Lie derivative of the Hodge dual $4$-form of $\varphi$.In terms of this decomposition, we characterise the conditions for a vectorfield as an infinitesimal symmetry of a coclosed $G_2$-structure, as well asthe soliton condition for the Laplacian coflow. More specifically, we providean easier proof for the absence of compact shrinking solitons of the Laplaciancoflow. Moreover, we revisit the Laplacian coflow of coclosed $G_2$-structureson almost Abelian Lie groups addressed by Fino-Bagaglini (2018). However, ourapproach is based on the bracket flow point of view. Notably, by showing thatthe norm of the Lie bracket is strictly decreasing, we prove that we havelong-time existence for any coclosed Laplacian coflow solution.

On the Laplacian coflow of invariant $G_2$-structures and its solitons