Is it possible that "axis" and "axioma" (English "axiom") derive from the same etymological root?
From a superficial search, it at least seems possible that the PIE root *ag "to move" (i.e. axiom: to move about ~) could translate across. However, I could find no explicit evidence that they were related etymologically.
It is indeed possible that Latin axis (whence English axis) and Greek axiôma (whence English axiom) are from the same Proto-Indo-European root, *h2eg- "to lead".
Chantraine describes the development of Greek axiôma from the verb axioô:
About the ultimate etymology of the root, he suggests that it is rightly connected with the verb ago "to lead, bring, carry, weigh":
De Vaan gives *h2eks-(i-) "axle, arm-pit" as the Proto-Indo-European root of Latin axis:
He also adds that this root may be derived from the Proto-Indo-European root *h2eg- "to lead", which is the root of the Latin verb ago, which has in turn the same origin as the Greek verb agô that Chantraine suggests as the origin of the Greek verb above:
