Oxidation of self-duality to 12 dimensions and beyond

Using (partial) curvature flows and the transitive action of subgroups of O(d,Z) on the indices {1,...,d} of the components of the Yang-Mills curvature in an orthonormal basis, we obtain a nested system of equations in successively higher dimensions d, each implying the Yang-Mills equations on d-dimensional Riemannian manifolds possessing special geometric structures. This `matryoshka' of self-duality equations contains the familiar self-duality equations on Riemannian 4-folds as well as their generalisations on complex K\"ahler 3-folds and on 7- and 8-dimensional manifolds with G_2 and Spin(7) holonomy. The matryoshka allows enlargement (`oxidation') to a remarkable system in 12 dimensions invariant under Sp(3). There are hints that the underlying geometry is related to the sextonions, a six-dimensional algebra between the quaternions and octonions.