Analysis of information sources in references of the Wikipedia article "Skleros" in English language version.
the Sclerus family [...] was a mixture of Greek and Armenian with roots in Cappadocia.
The Skleros family is often touted as "Armenian," but the evidence for this claim is just as tenuous as for the Lakapenoi, in fact more so. It consists of one early ninth-century Skleros said by a highly unreliable text (the Chronicle of Monembasia) to have come from the Roman province of Armenia Minor (not that he was "an Armenian"), and this figure may or may not have been related to the later family with the same surname. No source claims that any later Skleros—say, in the later tenth century—was "an Armenian," though many scholars make that claim. [...] Let us keep in mind that Skleros' "Armenian roots" have not been proven, anyway.
The fact that the Sclerus family had a Greek surname (skleros: "hard", or "severe") indicates, however, that its founding father may have been at least partly Greek; Byzantines of purely Armenian origin generally had surnames that were recognizably Armenian with a Greek suffix. Of the immediate ancestry of Bardas Sclerus, little is known.