On Frobenius functionals of the Lie algebra M₃(ℝ) ⊕ gl₃(ℝ)

In the present paper, we study the Lie algebra written in the semi-direct sum formula of the vector space M₃ (ℝ) and the Lie algebra gl₃ (ℝ) whose both contain 3×3 real matrices. We denote it by g₃ : = M₃ (ℝ) ⊕ gl₃ (ℝ). The aim of this research is to study the existence of a linear functional such that g₃ is the Frobenius Lie algebra of dimension 18. Such the linear functional is called the Frobenius functional. We applied the literature reviews to achieve this result, particularly we study the notion of Frobenius Lie algebra in Ooms and Rais results. The main result of our research is the proof that g₃ is Frobenius Lie algebra. For the future research, the existence of a Frobenius functional is still an open problem to study for higher dimensional Lie algebras. (paper)