  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.
