Meant to be free of cultural preconceptions, but it uses 24-hour division.

There might be a natural, Schelling division to use -- the same ratio year:day could be applied, day:minute. Thus, the day can be broken up into 365.25 "Schelling-minutes" (corresponds to 3.94 SI minutes), and further into 365.25^2 "Schelling-seconds" (corresponds to 0.65 SI seconds). I think it's interesting that we get something not too distant from our own minutes and seconds if we do this.

Of course, for practical daily usage, lacking anything similar to an hour would be a problem. A "half-division" might be in order; that is, √365.25, or roughly 19.11 divisions of the day. Sure, that's close enough to SI hours... but it's probably worth remembering that there's a reason we don't like to use fractional divisions. :P