Abstract
In this research, we studied quasi-Frobenius Lie algebras and filiform Lie algebras of dimensions ≤ 5 over real field. The primary objective of this research is to classify the classification of filiform Lie algebras of dimensions ≤ 5 into quasi-Frobenius Lie algebras. The method employed in this research involves constructing a skew-symmetric 2-form in real Lie algebra, which also a nondegenerate 2-cocycle. The outcomes of this research reveal that there exists a class of filiform Lie algebras of dimensions $\le 5$ that can be classified as a quasi-Frobenius real Lie algebra. Furthermore, this research can be developed to classify higher dimensional filiform Lie algebras as quasi-Frobenius real Lie algebras.
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