Optimal system of non-similar subalgebras of sum of two ideals

We consider a twelve-dimensional Lie algebra $L_{12}$ admitted by the gas dynamic equations with state equation of a special form. Lie algebra $L_{12}$ is a direct sum of two ideals $L_{11}$ and $Y_1$. For Lie algebra $L_{11}$ admitted by gas dynamic equations with an arbitrary equation of state, the optimal system of non-similar subalgebras is built up to inner automorphisms. Using the optimal system for Lie algebra $L_{11}$, in the article we obtain an optimal system of non-similar subalgebras of the sum of two ideals for $L_{11}$ and $Y_1$ and the rule of construction of such subalgebras.