Start with the modern mevalonate pathway, from wiki.
Now, let's butcher it down to its utter basics:
Isopentenyl pyrophosphate is the central cornerstone of polyprenyl phosphate generation by this reaction:
You can make the polyprenyl phosphate lipid as long as you like, just add more isopentenyl derived units. This gives us the entry point, top centre, for this diagram:
Down the left hand side we have the simple pathway to the archeal plasma membranes, as you might expect. Down the centre is the route to cholesterol, essential to animal cell membranes. The hopanoids, essential to bacteria but absent from archea, are left of centre.
Archea stayed with polyprenyls for their membrane, bacteria went for the fatty acids while retaining polyprenyl derived components. Derivatives of polyprenyls are ubiquitous. Metabolic cornerstones common to both archea and bacteria may well have been present in LUCA. Seems a reasonable argument to me.
There are a set of core enzymes present in both archea and bacteria which appear to have evolved before their divergence from LUCA. Perhaps the most significant is the rotator/stator ATP synthase, a piece of molecular nanotechnology of breathtaking complexity, considering that it appears to have evolved as one of the core early molecular machines of life. It's association with membranes suggest that membranes were there quite early on in evolution.
The presence of membranes allows us to think about some interesting questions. Such as why are cells powered by ATP? Because that is what ATP synthase produces. There is nothing intrinsically special about ATP. Reduced ferredoxin is a common power source in bacteria and archaea. Acetyl phosphate can be used too. But ATP is what is made by the ATP synthase machine and it can be produced in bulk, given a membrane potential. The hows and whys are interesting.