  We observe that, for a Bismut Einstein metric, the (2,0)-part of Bismut Ricci
form is an eigenvector of the Chern Laplacian. With the help of this
observation, we prove that a Bismut Einstein metric with non-zero Einstein
constant is K\"ahler Einstein. Additionally, for Bismut Einstein metrics with
zero Einstein constant, we prove that they are actually Bismut Ricci flat.
