From the Torah Minim The Four

וּלְקַחְתֶּם לָכֶּם בַיּוֹם הָרִ אשׁוֹן פְרִ י עֵץ הָדָר כַפֹּת תְמָרִ ים וַעֲנַף עֵץ־עָבֹּת וְעַרְ בֵי־נָחַל וּשְמַחְתֶּם לִפְ נֵי ה' אֱלֹקֵיכֶּם שִׁבְעַת יָמִים: (ויקרא כג, מ)

Even though the actual identity and laws of the Four Minim were handed down by Oral law, Chaza”l wished to reveal the allusions found within the written Torah.

• וּלְקַחְתֶּם
  To be read as if written וּלְקַח תָם the taking must be complete. None of the Four Minim may be missing (need all four) and none of the individual items may be missing of mutilated (חסר ). (Sukkos 34b)
  Had the Torah written in the singular, one might have thought that the mitzvah was for Bais Din to do as a representative of the Jewish People. The plural “you” indicates that the mitzvah is upon each Jew. (Sukkos 41b)
• לָכֶּם
  The four Minim require legal ownership to the Jew; they cannot be stolen or borrowed. (Sukkos 41b)
  The four Minim have to be fully available and cannot be of the type that is prohibited to have benefit, such as Orlah, an Asheira, or Ir haNidachas. (Sukkos 29b)
• בַיּוֹם
  The Mitzvah is done by day and not be night. (Sukkos 43a)
  The entire day is proper for the Mitzvah to be done. (Megilah 20b)
• בַיּוֹם הָרִ אשׁוֹן
  Just as the verse later says “seven days” it means to the seven days of Sukkos, so too the first day means to the first day of Sukkos (and not to be interpreted as the day prior to Sukkos. (Pesachim 5a)
  On the first day of Sukkos, the mitzvah of the four Minim applies everywhere; for the rest of Sukkos the mitzvah only applies in the Bais haMikdash, After the destruction, the mitzvah was instituted everywhere for the seven days of Sukkos. (Sukkos 41a)
• פְרִ י עֵץ הָדָר
  Read as הִדֻר (beautiful) to indicate that any of the four Minim that lack “beauty” are disqualified. (By other Mitzvos it is an optimal action to beautify the Mitzvah; by the Four Minimum is a requirement. (Sukkos 35a)
  Only use one for the Mitzvah. Probably not from the word פרי which is also used for multiples. Perhaps the word הדר indicates singular. (Sukkos 35a)
  This term refers to a tree whose fruit and wood tastes alike showing beauty. (Sukkos 35a)
  As if to be read הדר that dwells on the tree from one year to the next year. (Sukkos 35a)
  As if to be read אִידוּר meaning water, referring to a tree that needs to be irrigated with a lot of water. This derivation seems to be from the Greek root. (Some say all languages are concealed in the Torah; others say that during the Dispersion many Hebrew words were incorporated into other languages.)
• כַפֹּת תְמָרִ ים
  Since the pasuk does not place a “vav” prior to the phase, we only take one of this item (just like the previous item). (Sukkos 32a)
  The word כפת is read as plural but can be read as singular. Therefore, this indicates a type of plant that has one leaf that are really two (twin-leaf).
  The correct singular form of the word would be כף, the word כפת can be read as כָפֻת (tied up) indicating that this item can be tied up if it did spread apart. (Sukkos 32a)
• וַעֲנַף עֵץ־עָבֹּת
  The term literally means “shoot of the plaited tree” with ענף as the twig taken off the tree. However, Chaza”l understood the word ענף as part of the description and thus reads “the plaited-shoot tree.” This indicates that the tree does not have regular branches visible but seems to be full of shoots (leaves) growing in a plaited pattern, hugging the branches, totally covering them. (Sukkos 32b)
  Three of these min are taken; one for each word in the verse. (Sukkos 34b)
• וְעַרְ בֵי־נָחַל
  ערבי indicates two separate requirements to take the aravos; one to bundle with the lulav; and the second to encircle the Mizbeach each day in the Mikdash. (Sukkah 34a)
  The preferred mitzvah is to take stems of a willow that actually grow besides a brook. However, any stem of the willow species is equally fit for the mitzvah. (Sukkos 33b)
• וּשְמַחְתֶּם
  Even though every Yom Tov has a mitzvah to be happy, Sukkos in the Mikdash contains an extra happiness. (Rambam)
• שִׁבְעַת יָמִים
  The part of the mitzvah that lasted for seven days only applied when “before HaShem” in the Bais haMikdash (according to the Rambam in Yerushalayim). (Sukkos 41a)
  As explained the mitzvah of the Four Minim from the Torah only applies to the first day outside of the Mikdash. Thus, Chaza”l relaxed some of the requirements for the last six days of Sukkos. Neither the requirement of a “complete species” or of ownership applies after the first day of Sukkos. The requirement of “beauty” for the last six days is a matter of dispute. (Sukkos 36b)

Conditions to invalidate the Four Minim fall under three categories, derived from the three words in the Pasuk:

1. וּלְקַחְתֶּם None of the species may be absent and nor part of the individual species may be missing or damaged
2. לָכֶּם The mitzvah requires ownership
3. הָדָר No species may lacks the halachic standard of “beauty”

Baal haTurim:

• פרי עץ הדר same Gematria (659) of אתרוגים (within one)
• עבת (plaited) lacking the letter vav indicating 6 items invalidate the Hadasim: dried out; stolen, from an Ashirah tree, from a subverted city (Ir nidachas), its tip cut off, or associated with idolatry.
• Lulav לולב same Gematria (68) as חיים; and thus dried-out would be invalid.