Starting from $G_{2}$, the road toward $E_{n}$ goes via octonion algebras, triality, exceptional Jordan and structurable algebras, and other exceptional groups. This road is particularly non-trivial when the groups are studied, not over fields, but over commutative rings, whether those encode discrete structures or high-dimensional geometry. In this talk, I will follow this road, commenting on recent results where, starting from $G_{2}$, we combine torsors and non-associative algebras to study groups of type $D_{4}$, $F_{4}$, $E_{6}$ and $E_{7}$.