Commentaire sur Zevahim 1:4

קבל והלך וזרק – either or is taught. When he slaughtered it not for their own sake, or received he blood in a bowl out of which the sprinkling is done, or he brought the blood to the [golden] altar, or sprinkled/cast it. Each of these four [acts] of Divine service that he did not for their sake, with regard to the Passover offering or with the sin-offering, he has invalidated. But with the rest of the offerings, they did not count for the owners for the sake of their obligation, even if he did one of these four [acts] of Divine service for their own skae and not for their own sake together, whether he advanced the thought for their own sake to the thought not for their own sake, whether that he advanced the thought not for their own sake to the thought for their own sake, in every matter, it is invalid.

A pesah and a hatat which were slaughtered not in their own name, or he received [the blood], and carried it [to the altar] and sprinkled [it] not in their own name,
Or in their own name and not in their own name, or not in their own name and in their own name, they are disqualified.
What is the case of ‘in their own name and not in their own name’? In the name of it being a pesah [first] and [then] in the name of it being a shelamim.
‘Not in their own name and in their own name:’ in the name of a shelamim [first] and [then] in the name of a pesah.
For a sacrifice can be disqualified in [any one of] the four elements: slaughtering, receiving, carrying and sprinkling.
Rabbi Shimon declares it valid if carried [with the wrong intent], for Rabbi Shimon said: it is impossible [to have a valid sacrifice] without slaughtering, without receiving and without sprinkling, but it is possible without carrying. [How so]? One slaughters it at the side of the altar and sprinkles.
Rabbi Elazar says: if one goes where he needs to go, an [illegitimate] intention disqualifies [it]; where he doesn’t need to go, an [illegitimate] intention does not disqualify [it].
Section one: This section expands upon the halakhah that we learned in mishnah one, that a pesah or a hatat that were offered with the intent of their being a different sacrifice are disqualified. Here we learn two new halakhot. First of all, if any of the other essential parts of the sacrifice are done with the intent of the sacrifice being something else other than a pesah or a hatat, the sacrifice is invalid. The four essential elements of sacrifices are: slaughtering, receiving the blood, carrying it to the altar and sprinkling it on the altar.
Sections 2-4: The second new law we learn is that if one of these actions is done with the proper intent, but another of the actions is done with the improper intent, the sacrifice is invalid. The mishnah now illustrates this. If the sacrifice is supposed to be a pesah and it is first done in the name of it being a pesah and then later on the priest is confused and performs one of the later actions with the intent of it being a shelamim, the sacrifice is invalid. The same is true if at first he has the wrong intention and then when performing one of the later actions he has the correct intention. In either case the sacrifice is disqualified.
Section five: This supports what was stated above in section one any of the four essential aspects of the sacrifice can also serve as potential disqualifiers, if the intention is incorrect.
Section six: Rabbi Shimon disagrees with the statement in section five and holds that if the sacrifice’s blood is carried to the altar with the wrong intent, it is not disqualified because carrying is not essential to all sacrifices. Theoretically, one could slaughter a sacrifice right next to the altar and then sprinkle the blood without having to carry the blood from the point of slaughter to the altar.
Section seven: If the person is carrying the blood from the point at which he slaughtered it to the altar and while doing so he has the intention that it should be a different sacrifice, then the sacrifice is disqualified. However, if he slaughtered it next to the altar and received the blood in a vessel right there and then brought the blood elsewhere, an action that he did not need to do, and while carrying it he intended to offer it as another sacrifice, then the sacrifice is not disqualified, at least according to Rabbi Elazar. This carrying of the blood was unnecessary and therefore his intents at that moment do not factor into determining the validity of the sacrifice.

אבל אפשר שלא בהלוך – therefore, it is not considered to become ineligible. But the Sages state, that evefn though it is possible without conveying [the blood], one’s thought invalidates it. For Divine service which is possible to nullify is called Divine service, and conveying the blood is Divine Service, which is invalid with a foreigner (i.e., a non-Kohen), as it is written (Leviticus 1:5: “[The bull] shall be slaughtered [before the LORD;] and Aaron’s sons, the priests, shall offer [the blood against all sides of the altar which is at the entrance of the Tent of Meeting],” from reception and onwards is the command of the priesthood.

המהלך במקום שהוא צריך להוליך – this conveying, thought invalids it. How so? They received it [the blood] outside there from the altar and brought it inside, that they came close to the side of the altar. This is the conveying that needs to be done, and the thought/intention invalidates it. If they received it inside near the altar and brought it outside, that is a conveying that is unnecessary and thought/intention does not invalidate it. If he then went back and brought it inside, this is the conveying that is needed. But the Halakha is not according to either Rabbi Eleazar nor according to Rabbi Shimon.