Mishnayos Sotah Perek 3 Mishnah 1

היה נוטל – her husband [would take] her meal-offering, for the command of the priesthood regarding meal-offering is from taking a fistful of the meal-offering and onward (see also Talmud Menahot 9a) and bringing near [the meal-offering] is also the beginning of taking a fistful of the meal-offering but a foreigner (i.e., non-Kohen) does not make an offering on the altar but the sanctification of the utensils and the pouring of oil (on the flour – see Leviticus 2:1) and the mixing, as meal-offerings require oil are appropriate for non-Kohanim.

וכהן מניח ידו תחתיה – as it states regarding peace offerings (Leviticus 7:30): “his own hands shall present [the LORD’s offerings by fire],” and it is stated regarding the meal-offering of the woman suspected of unfaithfulness by her husband (Numbers 5:25): “Then the priest shall take from the woman’s hand [the meal-offering of jealousy,” just as with peace-offerings, the Kohen places his hand underneath the hand of the owners and waves it, so also in the meal-offering of the woman suspected of unfaithfulness by her husband, the Kohen places his hand underneath the hand of the woman and waves it.