G${}_2$-instantons on $2$-step nilpotent Lie groups

We study the G${}_2$-instanton condition for a family of metric connections arisen from the characteristic connection, on $7$-dimensional $2$-step nilpotent Lie groups with left-invariant coclosed G${}_2$-structures. According to the dimension of the commutator subgroup, we establish necessary and sufficient conditions for the connection to be an instanton, in terms of the torsion of the G${}_2$-structure, the torsion of the connection and the Lie group structure.Moreover, we show that in our setup, G${}_2$-instantons define a naturally reductive structure on the simply connected $2$-step nilpotent Lie group with left-invariant Riemannian metric. Taking quotient by lattices, one obtains G${}_2$-instantons on compact nilmanifolds.