Things That Invalidate The Four Species, 6 Seifim: 1. All of the four species are rendered invalid if they are obtained by theft or robbery, whether the owners have not yet given up hope of recovering or whether they have. However if one stole it and then established ownership of it by means other than doing the mitzvah, like one who stole a lulav and then improved it, it is valid, as he acquired it with this changing action. However, he should not bless on it. There is one who says that the object obtained by theft or robbery is only invalid for the thief or robber themselves, but is valid for others on days besides the first day. RAMA: Because of this one should be careful not to have a Jew themselves trim one of the [trees of the] four species of the lulav to make a lulav, because land cannot be [acquired by a changing action after it was] stolen and the typical non-Jew steals land, meaning that the stolen object will end up in one's hands. Rather, have the non-Jew trim and buy it from him (responsa of the Rashb"a 852, Hagahot Ashiri end of ch. "Lulav haGazul"). There is no distinction in this regard between the Land of Israel and outside of the Land (Or Zarua). A lulav that was made and assembled by a non-Jew is valid, just like the sukkah of a non-Jew (Mordechai in Minor Halakhot).
2. The same applies to a borrowed [lulav] on the first day, when we need it to be "yours." RAMA: One who takes a vow not to benefit from his lulav or from another's lulav cannot fulfill his obligation with it on the first day, because it is not "theirs" (responsa of Rashb"a 746 and 747).
3. Similarly, [a lulav] from a condemned city [due to idolatry] or one from an Ashera tree belonging to a Jew is invalid. However, one belonging to a non-Jew should not be taken initially, but if one took it on a day after the first day, they fulfilled their obligation, as we do not require it to be "yours." RAMA: This is specifically when one does not intend to acquire it. But if one intends to acquire it, it is a Jew's, and the obligation is not fulfilled. This is specifically before it is nullified [of its idolatry], but if it was nullified by the non-Jew, even if one intends to acquire it afterward, the obligation is fulfilled, though not initially (Rabbeinu Yerucham 8:3 and Ra"n). See 586.
4. One is permitted to take a lulav or other of the species for the mitzvah from the gardens or similar things at their places of worship [even if the tree was planted in front of idol worship as long as they have not worshiped the tree) (Rabbeinu Yerucham 8:3)].
5. Whenever we said that they were invalid due to defects or because they were stolen or robbed, this only applies on the first day of the holiday, but they are valid on the other days. RAMA: Some prohibit the stolen [lulav] for all seven days, and this is the practice. But one can fulfill the obligation with a borrowed [lulav] (Tur and Maggid in the name of the Poskim). It is permitted to take an acquaintance's lulav without permission the other days, because people like when a mitzvah is done with their possessions, and it is considered borrowed (Trumat Hadeshen 100 and his responsa 159). One with part of it missing is valid the rest of the days (Tur). One missing its pitom or the bottom tip is treated like one with part missing and is valid after the first day (Rabbeinu Yerucham 8:3). However, if mice punctured it, it should not be taken even on other days because it is gross (Kol Bo) until the mice bites are removed. However, if it was dry or spotted it is invalid all seven days even if the dry spot or spots are cut off, because it came from something invalid (Hagahot Ashiri ch. "Lulav haGazul"). It is permitted, initially, to stipulate on an etrog that it be just for the first day and that one will not separate from it from twilight of the second day on, and then it is permitted to eat from it on the second day on and to use it to fulfill the mitzvah if enough remains (Beit Yosef quoting the Ros"h), though we are not experts in stipulations as we said above at the end of Siman 638 in the note. If it is invalid because of idolatry, because it is forbidden to eat the etrog, because it is not the right species, or because it is not the right size, it is invalid both on the first day and on the subsequent days (and boils invalidate all seven days (Rabbeinu Yerucham 8:3 and Ra"n ch. "Lulav haGazul").) Those who make two days of Yom Tov can take things that are invalid on the first day and use them on the second day but should not bless on them (and if somebody else has a valid lulav and etrog, they should bless on the other one with permission (his own opinion)).
6. In an emergency when there is no valid set, any invalid set should be taken without blessing. RAMA: Some permit even blessing on a dry lulav (Rambam ch. 8 and the Maggid) and the practice is to bless on dry lulavs even when there are other wet ones (Agur). However, one does not behave that way with the other species. Some are lenient even with dry myrtle (Hagahot Maimoni and Mordechai), and they can be relied on in an emergency. Even though an etrog missing a part is valid on the later days, one should not cut up an etrog into two or three pieces and distribute it to fulfill the obligation, even in an emergency, because only one missing a part that still has the core is valid, but in this case, it would be considered slices of an etrog and not an etrog (Ra"n and Piskei Mahara"i 52). All of this refers to the blessing. Without a blessing, one can take any of the invalid ones and not bless on them (Tur).