On homogeneous closed gradient Laplacian solitons

We prove a structure theorem for homogeneous closed gradient Laplacian solitons and use it to show some examples of closed Laplacian solitons cannot be made gradient. More specifically, we show that the Laplacian solitons on nilpotent Lie groups found by Nicolini are not gradient up to homothetic $G_2$-structures except for $R^7$, where the potential function must be of a certain form. We also show that one of the closed $G_2$-structures constructed by Fernández-Fino-Manero cannot be a gradient soliton. We then examine the structure of almost abelian solvmanifolds admitting closed non-torsion-free gradient Laplacian solitons.