Balanced Hermitian structures on twisted cartesian products

We study Hermitian structures on twisted cartesian products $({\mathfrak{g}}_{({\rho }_1,{\rho }_2)},\mathrm{J},\mathcal{K})$ of two Hermitian Lie algebras according to two representations ${\rho }_1$ and ${\rho }_2$. We give the conditions on $({\mathfrak{g}}_{({\rho }_1,{\rho }_2)},\mathrm{J},\mathcal{K})$ to be balanced and locally conformally balanced. As an application we classify six-dimensional balanced Hermitian twisted cartesian products Lie algebras.