We now know that embryophytes are a monophyletic group derived from within the charophycean green algae (Karol et al., 2001; Turmel et al., 2002).
Coleochaete, Chaetosphaeridium, stoneworts, and embryophytes all produce a multicellular haploid body, and all produce large non-motile 'female' gametes and small motile 'male' gametes.
Coleochaete and embryophytes retain their zygotes on the maternal gametophyte, whereas Chaetosphaeridium expels its ova before fertilization.
This implies that interpolation of a multicellular diploid phase, between syngamy and meiosis, evolved after the divergence of stoneworts and embryophytes. Other, less parsimonious interpretations, are of course logical possibilities.
In summary, the last common ancestor of stoneworts and embryophytes can tentatively be proposed to have possessed a multicellular, oogamous, haploid gametophyte that did not produce zoospores (absent in Chara and embryophytes, present in Coleochaete).
For purposes of further discussion I will assume that Remy's reconstruction of the 'rhyniophyte' life cycle is correct, as are modern phylogenetic reconstructions in which embryophytes and tracheophytes are monophyletic, but 'bryophytes' are paraphyletic.
Two somewhat conflicting theories (the homologous theory and antithetic theory) of the origin of alternating generations, specifically the origin of the sporophyte, in embryophytes (land plants) have had respective supporters for approximately a century.
In any case, overwhelmingly, the sporophytes of embryophytes do not closely resemble the gametophytes on which they depend.
The larger flagellar root is a "band" with many microtubules (perhaps 60 or so) and is associated (toward the base) with a distinctive multilayered structure (MLS) composed of microtubules and laminate plates; this composite structure is similar to that found in sperm of embryophytes (cf.
(Chrorophyta and Charophyta) share the same types of chlorophylls (a and b) and carotenoids (e.g., lutein, beta-carotene) with embryophytes and, associatedly, similar chloroplast structure and thylakoid arrangement (Van den Hock et al., 1995).
Regardless, the Viridiplantae are considered by virtually all authors to represent generally related groups of organisms, some details not withstanding; in most cladistic analyses it seems clear that, among algae, charophytes place the closet to lower embryophytes.
The selection of the flattened-spheroidal, sometimes parenchymatous Coleochaetae as possibly representative of the putative ancestor of embryophytes (particularly thallose liverworts) is not a new idea and was clearly suggested by Bower (1908) and later supported by Campbell (1940).