A remark on Ricci flow of left invariant metrics

We prove that the Ricci flow equation for left invariant metrics on Lie groups reduces to a first order ordinary differential equation for a map $Q : (-a,a) \to UT$,where$UT$ is the group of upper triangular matrices. We decompose the matrix $R_{ij}$ of Ricci tensor coordinates with respect to an orthonormal frame field $E_i$ into a sum $\overset{1}{R}{ij} + \overset{2}{R}{ij} + \overset{3}{R}{ij} + \overset{4}{R}{ij}$ such that, for any $E_{i^{\mathrm{\prime }}}=U_{i^{\mathrm{\prime }}}^i{E}_i$ with $\mathrm{\mid }\mathrm{\mid }{U}_{i^{\mathrm{\prime }}}^i\mathrm{\mid }\mathrm{\mid }\in O(n)$, ${}_{i^{\mathrm{\prime }}{j}^{\mathrm{\prime }}}=U_{i^{\mathrm{\prime }}}^i{}_{ij}{U}_{j^{\mathrm{\prime }}}^j$. This allows us to specify several cases when the differential equation can be simplified. As an example we consider three-dimensional unimodular Lie groups.