The automorphism group of the octonions is the compact real form of \(G_2\).

The automorphism group of the split octonions is a noncompact real form of this group, the split real form. Let us call this \(G_2'\).

\(G_2'\) has an 8-dimensional representation on \(\mathbb{O}'\). But it preserves the 7-dimensional space of split octonions orthogonal to \(1 \in \mathbb{O}'\). These are the imaginary split octonions: $$ \mathrm{Im}(\mathbb{O}') = \big\{ (a i + b j + c k, \; d + e i + f j + g k) \big\} $$ where \(i,j,k\) are a basis of imaginary quaternions with $$ i^2 = j^2 = k^2 = ijk = -1 $$ and \(a,b,c,d,e,f,g \in \mathbb{R}\).