Classification of symbols of three-dimensional vector distributions of infinite type

We consider non-degenerate fundamental Lie algebras $\mathfrak{m}$ of infinite type over an arbitrary field of zero characteristic that can be uniquely represented as special extensions $0\to\mathfrak{a}\to\mathfrak{m}\to\mathfrak{n}\to0$, where all homogeneous components of $\mathfrak{a}$ are of dimension one. We provide explicit description of all such extensions in cases when $\mathfrak{n}$ is either a contact Lie algebra of dimension $\ge3$ or five-dimensional nilpotent Lie algebra of type $G_2$. In particular, get all fundamental Lie algebras $\mathfrak{m}$ of infinite type with $\dim\mathfrak{m}_{-1}=3$ and $\dim\mathfrak{n}\le5$. This covers all such Lie algebras $\mathfrak{m}$ that $\dim\mathfrak{m}\le 7$.