Who is "Adas Yisrael" referring to?
Targum Onkelos and Yonasan: It refers to K'lal Yisrael.
Sifra: Which we learn from a Gezeirah Shavah "Eidah" "Eidah" from Masaei, Bamidbar, 35:24-25 "Veshaftu ha'Eidah ... Vehitzilu ha'Eidah". And the same Gezeirah Shavah precludes where one member of the Sanhedrin is not fit to Pasken (Horayos, 4b). See Torah Temimah, note 41.
Sifra: Since the Torah adds the word "Adas Yisrael". See Sifsei Chachamim.
Why does the Torah add the word "Kol Adas Yisrael"
Horayos, 3b: To teach us that their Hora'ah only implicates them if the entire Sanhedrin of seventy-one) issued the erroneous ruling. 1
See Torah Temimah, note 50.
What are the connotations of "ve'Ne'elam Davar"?
Rashi: It means that the Halachah was hidden from them and they issued an incorrect ruling - to permit (be'Shogeg) something that is subject to Kareis.
What are the implications of the double Lashon "Yishgu Vene'elam"?
Horayos, 7b: It implies that the Sanhedrin issued an erroneous ruling and that Yisrael actually acted upon is.
Why does the Torah insert the word "Vene'elam Davar me'Einei ha'Kahalr"?
Horayos, 4a: To teach us - via a Gezeirah Shavah "Davar" "Davar" from Shoftim ("Ki Yipalei Mincha Davar ... Lo Sasur min ha'Davar"), Devarim, 17:8 in connection with Zakein Mamrei ? that Par ha'Eidah only applies where the Sanhedrin permitted part of a Chiyuv Kareis, but not where they permitted it completely. 1
See Torah Temimah, note 55.
Who is "Einei ha'Kahal" referring to?
Why does the Pasuk insert the word "ha'Kahal" (with a 'Hey') four times in the Parshah of Par ha'Eidah?
Horayos, 5a: 1. for itself; 2. to obligate each tribe (which is called "Kahal") to bring a bull, 1 3. to teach us that, although the ruling was issued by the Sanhedrin, the people must have acted upon it; 4. to teach us 'Gereirah' - that as long as either the majority of the people 2 or the majority of the tribles 3 sinned (even though the minority did not) they are obligated to bring twelve bulls, one for each tribe.
Who subsequently sinned?
Rashi and Seforno: The Kahal sinned (even if the Sanhedrin themselves did not. 1
Horayos, 5a: Because "ha'Kahal" implies the Sanhedrin - who issued the ruling, and "Ve'asu", the people - who carried it out.
Why does the Pasuk add (the otherwise superfuous) the word "Ve'asheimu"?
Seforno (on Pasuk 21): It is a warning to do Teshuvah before bringing the Korban.
Horayos, 8b: To preclude, via a Gezeirah Shavah "Ve'asheimu" "Ve'asheim" from the Chatas Yachid in Pasuk 27, and the Asham, in 5:17, that the Par ha'Eidah is confined to a 'Chatas Kevu'ah ? on a sin that carries a Chiyuv Kareis, but does not apply to a plain Asei and a Lo Sa'Asei in the Mikdash. 1
Sifra: To teach us that a Tzibur is subject to punishment just like a Yachid.
See Torah Temimah, note 62.