I didn't believe it was (since it was clearly using the product of two objects) and as such I was puzzled as to why you'd repeatedly post it in response to a request that asked about something else entirely.
nope, any 'natural' example, if any one knows of one. it is purely a 'for my own knowledge'
Ie is there an faithful representation of any (finite dimensional) lie algebra (any semisimple lie algebra certainly does)? Even if there is (which is eminently reasonablem though off the top of my head I don't know either) that wouldn't preclude it being a natural example of a space without composition, cf my unnatural example of the span of e,f,h = C^3 giving a copy of sl_2. There is no composition of elements e,f,h in C^3 given, though I am cheating by using a 'forgetful' construction.
Of course 'natural' is an undefined term.