Complex and bi-Hermitian structures on four-dimensional real Lie algebras

We give a new method for calculation of complex and bi-Hermitian structures on low-dimensional real Lie algebras. In this method, using non-coordinate basis, we first transform the Nijenhuis tensor field and bi-Hermitian structure relations on Lie groups to the tensor relations on their Lie algebras. Then we use adjoint representation for writing these relations in the matrix form; in this manner by solving these matrix relations and using automorphism groups of four-dimensional real Lie algebras we obtain and classify all complex and bi-Hermitian structures on four-dimensional real Lie algebras.