[jkqwzsoo]

> Do we know for a fact that all life derives from one organism? Could the conditions that allowed it to come into existence not have occurred in multiple places?

In the sense that we have directly observed this organism and can directly trace all life to it? No.

However, the evidence for the LCA is pretty strong. All doubts in my mind were removed when I took molecular genetics of procaryotes in college. Without going into details of finding relevant citations, and hence a fairly lay (and several years divorced from me learning about this stuff), the fact that all domains of life (procaryota, eucaryota, and archaea) share common genetic code and operate similar RNA- and protein-based molecular machinery strongly implies that a single "master template" originated all subsequent life on the planet. Considering how complex a ribosome is, for instance, the fact that all domains of life have similar, but not identical, ribosomal structures seems exceedingly unlikely. Molecular machinery is as complicated as it is because it evolves randomly to fulfill specific tasks. It's highly unlikely that two organisms would evolve the exact same piece of machinery which similar construction and composition to solve a novel task. On the other hand, when evolving from common machinery (i.e. shared ancestors), it's feasible to imagine that two organisms sharing an LCA might independently evolve similar proteins to adapt to a common stimulus. Consequently, eucaryotic cells function broadly similarly to other eucaryotic cells, archaeal cells function broadly similarly to other archaeal cells, and prokaryotic cells function broadly similar to other prokaryotic cells. Finally, archaeal cells function more closely to eucaryotic cells than do prokaryotic cells. Thus, it implies that archaea and eucaryota are more closely related than prokaryota and eucaryota. Hence, the phylogenetic tree. This implies that the most elegant (but not necessarily correct) solution to the similarity between different domains of life is a single unicellular ancestor.