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The set of all complex lines of the right-handed Dirac spinor bundle of a standard sixsphere is the total space of the twistor fibration. The twistor space, endowed with its natural Kahler structure, is recognised to be a six-dimensional complex quadric. The relevant group is Spin (7), which acts transitively on the six-quadric, as a group of fiberpreserving isometries. We use a result due to Berard-Bergery and Matsuzawa to show the existence of a non-Kahler, non symmetric, Hermitian-Einsteindoi:10.1017/s0004972700028021 fatcat:uswuvczqhbfwroegcuetcc7xle