Balanced Hermitian structures on twisted cartesian products

We study Hermitian structures on twisted cartesian products $(\mathfrak{g}_{(\rho_{1},\rho_{2})},\mathrm{J},\cal{K})$ of two Hermitian Lie algebras according to two representations $\rho_{1}$ and $\rho_{2}$. We give the conditions on $(\mathfrak{g}_{(\rho_{1},\rho_{2})},\mathrm{J},\cal{K})$ to be balanced and locally conformally balanced. As an application we classify six-dimensional balanced Hermitian twisted cartesian products Lie algebras.