פירוש על פרה 4:3

שרפה שלא בעצים – that he kindled the fire with the body of the cow/heifer.

If he burned it without wood, or with any kind of wood, and even if only with straw or stubble, it is valid. Although the type of wood that should be used to burn the red cow is listed in 3:8, if he uses other wood, or even no wood at all, the act is still valid. It seems that the wood is not part of the mitzvah, but just something that enables the mitzvah and therefore using different material does not invalidate the mitzvah.

כשרה – that “wood” is not written in the Torah, but [only] "ושרף"/”and he shall burn [the cow in his sight]” (Numbers 19:5) plainly/merely.

If he flayed it and cut up, it is valid. The act is valid even if the cow is flayed and cut up before being burned. Of course, all of the cow must be burned.

או בכל עצים – that he didn’t burn it (i.e., the cow) with cedar wood, pine and cypress and wood of fig trees that are taught [in the Mishnah] in the chapter above (see Tractate Parah, Chapter 3, Mishnah 8).

If he slaughtered it with the intention of eating its flesh or drinking its blood, it is valid. Rabbi Eliezer rules: no intention can invalidate the red cow. The printed version reads "valid" but this seems to be an erroneous reading. Manuscripts and testimony from medieval commentaries read "invalid" which makes more sense. If he slaughters the red cow with the intention of eating its flesh or drinking its blood, he has invalidated it. Rabbi Eliezer is consistent with his position in mishnah one wrong intentions do not invalidate the red cow ritual since it is not considered to be a sacrifice.

הפשיטה ונתחה כשרה (if he stripped/flayed it and dissected it) – it is written (Numbers 19:5): “[The cow shall be burned in its sight -] its hide, flesh [and blood shall be burned],” whether through flaying and dissecting it or whether attached.

על מנת לאכול מבשרה או לשתות מדמה כשרה – and specifically to eat and to drink that the first Tanna/teacher validates it, that with all of the sacrifices also in that manner it is kosher/fit, as it is taught in the Mishnah in the [Third] Chapter “All the Invalid”/כל הפסולים (Mishnah 3, [folio 35a], he thought about eating something that he generally doesn’t eat, it is kosher/fit, as for example, from the food of the Altar for a person, but if he thought on the condition to sprinkle from its blood on the morrow, the first Tanna/teacher invalidates, for the All-Merciful calls it a “sin-offering.” But Rabbi Eliezer holds that thought does not invalidate with regard to a cow at all, even with sprinklings. But he goes according to his reasoning that what is done not for its own sake he validates at the beginning of our chapter (see Chapter 4, Mishnah 1), which has no law from the laws of Holy Things, since it was performed outside [the Temple walls]. But the Halakha is not according to Rabbi Eliezer.