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Given an d-dimensional manifold with two commuting Killing vectors, togetherwith an d - 1 dimensional submanifold in which one of the Killing vectors lies,then the lapse and shift of the second Killing vector, relative to this slice,remain constant along the orbits of the `surface' Killing vector.Alternatively, the six dot products that can be formed from the three vectors,the two Killing vectors and the normal to the submanifold, are all constantsalong the `surface' Killing vector.

With commuting Killing vectors, the lapse and shift of one Killing vector are constants along the other