The eukaryotes share most of their metabolic paths with bacteria instead of archaea, not only their membrane composition.
Because all the known eukaryotes have their origin after the symbiosis with the bacteria that have become mitochondria and most of the genome of the mitochondria has been moved to the nucleus, one possibility is that all the bacterial characteristics of the eukaryotes come from the ancestor of mitochondria.
There are alternative hypotheses, e.g. that there have been a few other symbiosis events with some bacteria, before the symbiosis with the ancestor of mitochondria, so some of the bacterial characteristics are inherited from different bacteria.
A possibility is that some of the differences between archaea and bacteria have arisen after the divergence between archaea and eukaryotes, so the bacterial characteristics of the eukaryotes would be primitive. However this supposition has become more unlikely during recent years, because some evidence has accumulated that appears to support a closer relationship between eukaryotes and some of the archaea than with the other archaea, in which case all characteristics shared by all archaea would predate the divergence with the eukaryotes.
Because all the known eukaryotes have their origin after the symbiosis with the bacteria that have become mitochondria and most of the genome of the mitochondria has been moved to the nucleus, one possibility is that all the bacterial characteristics of the eukaryotes come from the ancestor of mitochondria.
There are alternative hypotheses, e.g. that there have been a few other symbiosis events with some bacteria, before the symbiosis with the ancestor of mitochondria, so some of the bacterial characteristics are inherited from different bacteria.
A possibility is that some of the differences between archaea and bacteria have arisen after the divergence between archaea and eukaryotes, so the bacterial characteristics of the eukaryotes would be primitive. However this supposition has become more unlikely during recent years, because some evidence has accumulated that appears to support a closer relationship between eukaryotes and some of the archaea than with the other archaea, in which case all characteristics shared by all archaea would predate the divergence with the eukaryotes.