Shevuot (Oaths) 3 - Who Authored the "Two that are Four" Ruling?
That ruling that teaches the increased liability could not be the opinion either of Rabbi Ishmael, nor of Rabbi Akiva. It could not belong to Rabbi Ishmael, because he says that one can be liable only for oaths that concern the future, not ones that are about the past.
Why could it not be the opinion of Rabbi Akiva? The other part of the "two that are four" ruling says this: "There are two types of awareness of impurity that are really four. One can forget that he is impure and eat sacrifices or enter the Temple - these are two. The other two are when one remembers that he is impure, but forgets that these are sacrifices that he is eating, or that it is the Temple area that he is entering." We know however that Rabbi Akiva does not make one liable for the second type of unawareness.
We could still say that the ruling is in agreement with both Rabbi Ismael and Rabbi Akiva, but that for the other two transgressions one is not liable to bring a sacrifice, but only to receive lashes.
Art: Isidor Kaufmann - Discussing The Talmud