We will show the classification of 6-dimensional unimodular Lie algebras ${\mathfrak g}$ that admit a complex structure with non-zero closed (3,0)-form.
This gives rise to compact quotients $M=\Gamma\backslash G$ endowed with invariant complex structures whose canonical bundle is holomorphically trivial.
In the balanced setting, we focus on the solutions of the Hull-Strominger system with respect to the $(\varepsilon,\rho)$-plane of metric connections $\nabla^{\varepsilon,\rho}$ that contains Levi-Civita and the Gauduchon line of Hermitian connections.
When $\nabla^{\varepsilon,\rho}$ is Hermitian-Yang-Mills, there are only three possibilities for ${\mathfrak g}$: the nilpotent ${\mathfrak n}_3$, the solvable ${\mathfrak s}_7$, and the semi-simple ${\mathfrak{so}}(3,\!1)$.