The Shulchan Aruch · Orach Chayim
אורח חיים תרנ״חOrach Chayim 658
Siman 658 of 697 in Orach Chayim — 9 se'ifim
הלכות לולבLaws of the Four Species· siman 658
Se’if 1א׳
מִן הַתּוֹרָה אֵין מִצְוַת לוּלָב חוּץ לַמִּקְדָּשׁ אֶלָּא יוֹם רִאשׁוֹן, וַחֲכָמִים תִּקְּנוּ שֶׁיְּהֵא נִטָּל בְּכָל מָקוֹם כָּל שִׁבְעָה.
The Laws of the Lulav on the First Day of Yom Tov. Containing 9 paragraph: By Torah law, the mitzvah of lulav only applies on the first day outside of the Temple, but the Sages decreed that one should do it everywhere for all seven days.
Se’if 2ב׳
בְּיוֹם שַׁבָּת אֵינוֹ נִטָּל, אֲפִלּוּ אִם הוּא יוֹם רִאשׁוֹן. הַגָּה: וְאָסוּר לְטַלְטֵל הַלּוּלָב בְּשַׁבָּת, דְּהָוֵי כְּאֶבֶן (רַ"ן סוֹף פֶּרֶק לוּלָב הַגָּזוּל וְהַמַּגִּיד פ"ז וְכָל בּוֹ); אֲבָל הָאֶתְרוֹג מֻתָּר בְּטִלְטוּל, דְּרָאוּי לְהָרִיחַ בּוֹ; וְאָסוּר לִתְּנוֹ עַל הַבֶּגֶד, אֲפִלּוּ בְּיוֹם טוֹב, דְּמוֹלִיד רֵיחָא (מַהֲרִי"ל).
It is not taken on Shabbat even if it is the first day. RAMA: It is forbidden to touch the lulav on Shabbat as it is likened to a stone (Ra"n the end of ch. "Lulav haGazul", Maggid ch. 4, Kol Bo). However, it is permitted to touch the etrog because it can be smelled. It is forbidden to put it on clothes even on Yom Tov because it imparts a scent (Mahari"l).
Se’if 3ג׳
אֵין אָדָם יוֹצֵא יְדֵי חוֹבָתוֹ בְּיוֹם רִאשׁוֹן. בְּלוּלָב שֶׁל חֲבֵרוֹ שֶׁהִשְׁאִילוֹ, דְּבָעֵינָן לָכֶם (וַיִּקְרָא כג, מ), מִשֶּׁלָּכֶם, וַאֲפִלּוּ אָמַר לוֹ: יְהֵא שֶׁלְּךָ עַד שֶׁתֵּצֵא בּוֹ וְאַחַר כָּךְ יְהֵא שֶׁלִּי כְּבַתְּחִלָּה, לֹא יָצָא, דְּהָוֵי כְּמוֹ שָׁאוּל; וְאִם נְתָנוֹ לוֹ בְּמַתָּנָה, מֻתָּר.
A person does not fulfill the obligation on the first day with a lulav loaned by somebody else because we require it to be "yours," of your possessions. Even if they said "This is yours until you fulfill your obligation, after which point it will be mine as it was initially," the obligation is not fulfilled because it is basically borrowed. However, if it is given as a gift, it is permitted.
Shulchan Aruch, Orach Chayim — Torat Emet 363 edition, via Sefaria — public domain
English: Sefaria Community Translation — CC0