We give a construction which is Lie theoretic of anti-invariant Riemannian submersions from almost Hermitian manifolds, from quaternion manifolds, from para-Hermitian manifolds, from para-quaternion manifolds, and from octonian manifolds. This yields many compact Einstein examples.
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Wikipedia, https://en.wikipedia.org/wiki/octonian. https://en.wikipedia.org/wiki/octonian Wikipedia, https://en.wikipedia.org/wiki/octonian. https://en.wikipedia.org/wiki/octonian
