Politeia means the ways and reasons of a people to live together, in friendship creating the institutions of their city. The word polis (πόλις) is included in politeia (πολιτεία) and it starts politeia, their living together as a sacred communion, devoted to Gods.
Arendt says, that “the life of polis meant for them (the Greeks), a very special and freely chosen form of political organisation and in no way just some form of action necessary to keep people united in an orderly cohabitation.” Polis is a place for discussion about everything and crucially about the whole meaning of being alive, not just about laws.
I don't know if there exists an English word which contains the city (polis) and upon the city bases a word which means the whole life of the city. The word 'political' is Greek, yet much more narrow than what it is in its Greek meaning.
Of course polis and politeia is not the State, as Castoriadis has observed, if we translate Pericles' funeral oration with State instead of Polis, as German philologues have done, the text becomes almost a nazi text: where Pericles said that ‘we die for each other’ (this 'each other' is the Polis), the translation says: we die for the State.
I have seen somewhere the translation of politeia into polity. This word (which corresponds to civility) has the advantage of being ONE word, but we really need two:
citizens' life, as you suggest.
If the polis is a "citizens' community," then politeia is the order or condition of such a community: the citizens' communality or "Civil Consociation." If those citizens are free, then Politeia corresponds to Republic, which can be defined as the society of free [lordless, anarchic] men. (A republic does not have a government; it has an administration of the few COMMON concerns of the citizenry [the common defense and public works], and a jurisprudential and judicial system under the aegis of Jupiter/Reason.) A Politeia differs from a republic, if it has a legislative government and a HEliaia, a people's assembly for the public judgmement of crimes.
"Politeia" may still be translated as "republic" more or less in its etymological sense: res publica, the Public Concern, or the Commonwealth (as some American States are still called in the sense of a politically organized body).
"Polity" would be the best choice. It already has a place in the political science's jargon -in English at least. Sixteen century Anglican theologian Richard Hooker titled his monumental work "A Treatise of the Laws of Ecclesiastical Politie". Hooker uses "politie" referring to two realms of human life: the civil one and the ecclesiastical one, each incarnated in the civil commonwealth and in the ecclesiastical commonwealth, respectively. Since then in Anglican canon law the word "polity" refers to the way churches of the Anglican Communion have organised themselves as communities of believers. So I believe we are on the right path: polity would be the English translation for "politeia". (Rev. J.C. Martin Anglican Cathedral, Mexico City.)
I suggest the consultation of the Index Aristotelicus by Hermann Bonitz (ed), published by Walter de Gruyter; you can find all the meanings of this word (and all the relevant) as used by Aristotle, with referrence to the works and specific passages. I give a few ones, which I believe more meaningful:
But the constitution (πολιτεῖαι) is an order of those who live in a city. (Politics Γ 1275b38) - Since the constitution (πολιτεία) is, somehow, the life of the city (Politics Δ 1295a40) - It is, consequently, evident that all the constitutions (πολιτεῖαι) which aim to the common interest are right constitutions (πολιτεῖαι) since they correspond to absolut justice, while those which aim to the private interest of governants are wrong and constitute degenerations with respect to the rightful constitutions (ὀρθῶν πολιτειῶν); in effect, they are despotical, while the city is a community of free men. (Politics, Γ 1279a17ss)