New examples of shrinking Laplacian solitons

We give a one-parameter family of examples of shrinking Laplacian solitons, which are the second known solutions to the closed $G_2$-Laplacian flow with a finite-time singularity. The torsion forms and the Laplacian and Ricci operators of a large family of $G_2$-structures on different Lie groups are also studied. We apply these formulas to prove that, under a suitable extra condition, there is no closed eigenform for the Laplacian on such family.