Octonionic Triality, the Matrix Structure of g2, and Principal Bundle Moduli Spaces

We develop a matrix-theoretic framework for the natural embedding of the exceptional Lie algebra g2=Der(O) in so(8), use it to make constructive a recent existence result on octonionic triality, and derive geometric applications for moduli spaces of principal bundles. Specifically, the derivation condition for D^∈so(7) is reformulated as a homogeneous linear system in the 21 entries of D^, whose solution space is identified with g2=kerΨ, where Ψ:so(7)→Λ3R7* is the Lie derivative with respect to the associative 3-form φ on Im(O). It is proved that rankΨ=7, and an algorithm is given for computing an orthonormal basis of g2. The image ΨA^σ of the triality generator is computed for all triples, yielding six nonzero components and squared norm 12. As geometric applications, the map Ψ is globalized to a morphism of adjoint bundles, giving an intrinsic characterization of the G2-reductible locus in M(SO(7)). The orthogonal decomposition of so(8) globalizes to an explicit splitting of the adjoint bundle of any SO(8)-principal bundle admitting a G2-reduction. Finally, M(G2) is identified as a connected component of the triality fixed-point locus in M(Spin(8)), with an explicit description of the tangent and normal spaces in terms of the Lie-algebraic decomposition.