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We study the geometry induced on the local orbit spaces of Killing vectorfields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G={\rmSpin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms withone-dimensional fibres from $G_2$-manifolds to Einstein manifolds.

On $G_2$-manifolds and geometry in dimensions $6$ and $8$