In short: the group \(G_2'\) has a 7-dimensional representation on the space of imaginary split octonions, \(\mathrm{Im}(\mathbb{O}')\).

It preserves the quadratic form \(Q\), which has signature \((3,4)\) on this space: $$ Q(a i + b j + c k, \; d + e i + f j + g k) = a^2 + b^2 + c^2 - d^2 - e^2 - f^2 - g^2 $$ So, this group acts on the lightcone: $$ C = \{x \in \mathrm{Im}(\mathbb{O}') : \; Q(x) = 0 , \; x \ne 0 \} $$ which is 6-dimensional, and the projectivized lightcone: $$ PC = \frac{C}{\{x \sim c x : c \in \mathbb{R}\}} $$ which is 5-dimensional.