SAMS Subject Classification Number: 29

We look at the Lie algebras obtained by modifying the structure constants of the complex Lie algebra \(\mathfrak{g}_2\) via contractions preserving the fine \((\mathbb{Z}_2)^3\)-grading, \(\Gamma_{\mathfrak{g}_2}\).

To do so, we consider how we may use a \((\mathbb{Z}_2)^3\)-grading of the Octonions to construct the corresponding grading \(\Gamma_{\mathfrak{g}_2}\) on the smallest exceptional Lie algebra \(\mathfrak{g}_2,\) which may be realized as the derivations of the Octonions. We then see how the special properties of this grading allow for the classification of the Lie algebras, obtained from contractions of the grading \(\Gamma_{\mathfrak{g}_2},\) to be converted into a combinatorial problem involving the Fano plane, \(P(\mathbb{Z}_2^3),\) and collineations of its points. Finally, we investigate some properties (nilpotency, solvability, centre) of the obtained Lie algebras and how these properties relate to the grading.