We provide an independent proof of the classification of the maximal totally geodesic submanifolds of the symmetric spaces G 2 G₂ and G 2 / S O (4) G₂/ SO (4), jointly with very natural descriptions of all of these submanifolds. The description of the totally geodesic submanifolds of G 2 G₂ is in terms of (1) principal subalgebras of g 2 g₂ ; (2) stabilizers of nonzero points of R 7 R⁷ ; (3) stabilizers of associative subalgebras; (4) the set of order two elements in G 2 G₂ (and its translations). The space G 2 / S O (4) G₂/ SO (4) is identified with the set of associative subalgebras of R 7 R⁷ and its maximal totally geodesic submanifolds can be described as the associative subalgebras adapted to a fixed principal subalgebra, the associative subalgebras orthogonal to a fixed nonzero vector, the associative subalgebras containing a fixed nonzero vector, and the associative subalgebras intersecting both a fixed associative subalgebra and its orthogonal. A second description is included in terms of Grassmannians, the advantage of which is that the associated Lie triple systems are easily described in matrix form.
Draper et al. (2025) studied this question.