פירוש על סוכה 4:2

לולב שבעה – for the first day [of Sukkot] supersedes the Sabbath because it has the essential [principle] from the Torah outside of Jerusalem, as it is written (Leviticus 23:40): “ On the first day you shall take [the product of hadar trees, branches of palm trees, boughs of leafy trees, and willows of the brook, and you shall rejoice before the LORD your God seven days].”

“The lulav for seven.” How so? If the first day of the festival fell on Shabbat, the lulav [is taken for] seven days; on any other day, [it is taken] for six.
As we have learned before, outside of Jerusalem it is a mitzvah from the Torah to take the lulav only on the first day of the festival. Hence, if the first day of the festival falls on Shabbat, one still performs the mitzvah of the lulav. We saw this described above in 3:13. In such a case the lulav will be taken for seven days one day in which the mitzvah is “deoraita” from the Torah, and the rest of the days it is “derabbanan” from the rabbis.
However, if Shabbat falls on another day besides the first day of the festival, then since the mitzvah is only of rabbinic origin, it is not observed. In other words, the derabanan mitzvah of the lulav does not override the Shabbat. In such a case the mitzvah is observed for only six days.
Since the time of the Babylonian Talmud it has become customary not to take the lulav on Shabbat, even if it falls on the first day of the festival. However, this is not the custom reflected in the mishnah.

ושאר כל הימים – if the First Day [of Sukkot] occurs on the other days of the week, and it is found that Shabbat occurs during the Intermediate Days of the Festival, it does not supersede [the Sabbath], and even though, from the Torah, all seven [are observed] in the Temple, the Sages preserved their decree on the other days of the Festival [of Sukkot] since they lack the principle from the Torah outside of Jerusalem. But by law, even in our times, the Lulav would supersede the Sabbath on the First Day of the Festival [of Sukkot], but because we are not expert in the establishment of the month, and lest that it is not the first day of the Festival, therefore, it does not supersede the Sabbath from doubt. But the reason that they (i.e., the Sages) decreed to not take up the Lulav on the Sabbath, for in taking it up, there is no [violation] of work other than merely carrying, and it was decreed lest he take it in his hand and would walk to someone expert to learn the blessing and the order of the Wavings [of the Lulav] and cause to pass four cubits in the public domain, and that is the reason for the Shofar [which is prohibited to carry on Shabbat] and that is the reason for the Megillah [which is prohibited to carry on Shabbat].