The Wikipedia article on the exceptional Lie group $G_2$ has the following definition of the group:
$$G_2=\{g\in SO(7):g^*\varphi=\varphi, \varphi = \omega^{123} + \omega^{145} + \omega^{167} + \omega^{246} - \omega^{257} - \omega^{347} - \omega^{356}\}.$$
What is $\omega^{ijk}$, what does the asterisk on $g$ mean, and what is the action of $g^*$ on $\omega^{ijk}$?
The Lie group $G_2$ can be defined as the group of automorphisms of the Octonions $\Bbb Q_8$. This definition can be shown to be equivalent to the subgroup of $SO(7)$ that preserves the $3$-form $$ φ = dx_{123} + dx_{145} + dx_{167} + dx_{246} − dx_{257} − dx_{347} − dx_{356} $$ in $\Bbb R^7$, where $dx_{ijk} = dx_i ∧ dx_j ∧ dx_k$. The $\ast$ is the Hodge operator, see here.
