The Didache lists the primary mode of baptism as that done in "living water." Classical usage of the word group denotes vitality (Bultmann, 832). In general, we would expect the sign to be self-movement, as opposed to some sort of a mechanical movement (Bultmann, 833). This would indicate in classical usage an expectation of the existence of a reasonable soul and the ability of perception, though the level of sophistication may well differ for various living beings.
In Hellenistic thought, as in Classical thought, the ζωή concept indicated motion (Bultmann, 837). Stoicism would typically differentiate between a merely physical life and a life enlivened by spirit, virtue, or reason. These were the elements considered by the Stoics to make true life (Bultmann, 838).
Neo-Platonism, with its philosophical dualism, also considered life to imply the presence of a reasonable soul (Bultmann, 838).
Gnosticism, counter to Neo-Platonism, separated ζωή from other attributes, making it "the divine life" (Bultmann, 839). While life is a physical phenomenon, it is something which in itself is indestructible. As such, it is an element which is divine. Interestingly, it is considered "as a something which flows into the believer" (Bultmann, 841), a characteristic which could easily be identified with running water used in baptism or other ceremonial washings.
Old Testament views of life and death, according to von Rad, may have been strongly influenced by "ideas which arose in the great cultural centres and which went out from these to surrounding territories as developed conceptions" (von Rad, 843). Throughout the Old Testament life is understood as an intrinsic good. It is considered as precious and a great fortune. One sign of the Fall is a decline in the length of human lifespans. Inherent in the curse is hardship of life (von Rad, 844). A blessed life, or, for that matter, any life at all, has God as its source. This is mediated not through any sorts of rites and rituals, but through God's word, delivered to his people (von Rad, 845). In contrast, the life which is not rooted in God's Word is expected to be sad and short.
While an Old Testament view of life is relatively rich, von Rad sees the view of death as "unitary and constant even through many centuries of vigorous religious development" (von Rad, 846). After death, there is no hope of reunion or restoration. The dead are gathered in sheol, all together. In life, there is a relationship with God. Death, however, takes one out of that realm, into a place where there is no praise to God (von Rad, 847).
Despite this view of sheol, von Rad recognizes an additional element in the Old Testament understanding of death. God remains the ruler over a higher sphere. While our expectation may well be of sheol, yet God is still the deliverer of His people (von Rad, 848).
Though the narrative in TDNT about the group of words related to ζωή is interesting, it is clear that the article is unlikely to shed significant light on my initial topic of inquiry, an adequate interpretation of "living water" as used in the Didache's baptismal passages. I am leaving the remaining 25 pages of the article unread for now.