  We study Hermitian metrics whose Bismut connection $\nabla^B$ satisfies the
first Bianchi identity in relation to the SKT condition and the parallelism of
the torsion of the Bimut connection. We obtain a characterization of complex
surfaces admitting Hermitian metrics whose Bismut connection satisfy the first
Bianchi identity and the condition $R^B(x,y,z,w)=R^B(Jx,Jy,z,w)$, for every
tangent vectors $x,y,z,w$, in terms of Vaisman metrics. These conditions, also
called Bismut K\"ahler-like, have been recently studied in [D. Angella, A.
Otal, L. Ugarte, R. Villacampa, On Gauduchon connections with K\"ahler-like
curvature, to appear in Commun. Anal. Geom., arXiv:1809.02632 [math.DG]], [Q.
Zhao, F. Zheng, Strominger connection and pluriclosed metrics, arXiv:1904.06604
[math.DG]], [S. T. Yau, Q. Zhao, F. Zheng, On Strominger K\"ahler-like
manifolds with degenerate torsion, arXiv:1908.05322 [math.DG]]. Using the
characterization of SKT almost abelian Lie groups in [R. M. Arroyo, R.
Lafuente, The long-time behavior of the homogeneous pluriclosed flow, Proc.
London Math. Soc. (3), 119, (2019), 266-289], we construct new examples of
Hermitian manifolds satisfying the Bismut K\"ahler-like condition. Moreover, we
prove some results in relation to the pluriclosed flow on complex surfaces and
on almost abelian Lie groups. In particular, we show that, if the initial
metric has constant scalar curvature, then the pluriclosed flow preserves the
Vaisman condition on complex surfaces.
