In unital JB*-algebras, an extreme point u of the closed unit ball is unitary if and only if the set of extreme points e with ||u ± e|| ≤ √2 contains an isolated point.
Peralta, A survey on Tingley’s problem for operato r algebras, Acta Sci
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Metric characterisation of unitaries in JB$^*$-algebras
In unital JB*-algebras, an extreme point u of the closed unit ball is unitary if and only if the set of extreme points e with ||u ± e|| ≤ √2 contains an isolated point.