if we reanalyze turings final argument can we say that he is testing for a given representation being one that is a group, given a lie algebra and its generator?

he really says that g has subgroup h which inturn has a centralizer subgroup h1

see http://www.turingarchive.org/viewer/?id=133&title=29

see also k:

### Generalization to Borel sets

This distribution can be generalized to more complicated sets than intervals. If *S* is a Borel set of positive, finite measure, the uniform probability distribution on *S* can be specified by defining the pdf to be zero outside *S* and constantly equal to 1/*K* on *S*, where *K* is the Lebesgue measure of *S*.

from https://en.m.wikipedia.org/wiki/Uniform_distribution_(continuous)