Theorem. If and only if \(R = 3 \) or \(R = 1/3\), the following notions of 'line' are the same:
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The path in \(P C\) traced out by a spinorial ball of radius \(R\) rolling without slipping or twisting along a geodesic in the projective plane coming from a sphere of radius \(1\).
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The collection of all 1-dimensional null subalgebras of the split octonions that lie in some fixed 2-dimensional null subalgebra.
In these two cases, the group of all transformations of \( P C \) that map lines to lines is \( G_2'\).
