The name of the portion, ‘Shemini’, means ‘the Eighth’, referring to the eighth day after completing the building of the Holy Tabernacle.
Sefat Emet (https://dailyzohar.com/tzadikim/158-Rabbi-Yehudah-Aryeh-Leib-Alter-of-Ger) in his commentary to this portion says that the eighth connects Malchut to the spiritual roots. The roots of everything in existence is in Binah, which is the Eighth Sefira when counting from Malchut up.
On the eighth day Moses called Aaron to prepare him for his new duty as a High Priest. Before Moses could connect him to the spiritual roots in Binah and serve in the Holy Tabernacle, Aaron needed to cleanse himself from past sins.
The Zohar explains that Aaron wasn’t punished for his part in the sin of the Golden Calf with the rest of the Israelites because Moses prayed for him.
“וּבְאַהֲרֹן, הִתְאַנַּף יְהוָה מְאֹד–לְהַשְׁמִידוֹ; וָאֶתְפַּלֵּל גַּם-בְּעַד אַהֲרֹן, בָּעֵת הַהִוא”
“”YHVH was angry enough with Aaron to destroy him; so I also prayed for Aaron at the same time.”
Now before he could serve the Holy, he needed to bring a sacrifice to cleanse his sins.
“וַיֹּאמֶר אֶל-אַהֲרֹן, קַח-לְךָ עֵגֶל בֶּן-בָּקָר לְחַטָּאת וְאַיִל לְעֹלָה–תְּמִימִם; וְהַקְרֵב, לִפְנֵי יְהוָה.”
“and he said to Aaron, “Take for yourself a calf, a bull, for a sin offering and a ram for a burnt offering, both without defect, and offer them before YHVH”
Then Moses instructed Aaron to tell the Israelites to bring ‘peace offering’, which is another type of sacrifice with the purpose was to bring completion and peace.
“וְשׁוֹר וָאַיִל לִשְׁלָמִים, לִזְבֹּחַ לִפְנֵי יְהוָה, וּמִנְחָה, בְּלוּלָה בַשָּׁמֶן: כִּי הַיּוֹם, יְהוָה נִרְאָה אֲלֵיכֶם.”
“and an ox and a ram for peace offerings, to sacrifice before YHVH, and a grain offering mixed with oil; for today YHVH will appear to you.”
The eighth day was a day of complete unification of upper and lower and everyone was to be cleansed from sins and in harmony with the Light that was revealed.
We can’t get away from our individual sins, even if a tzadik prays for us. We still need to make corrections and cleanse ourselves. Our spiritual work is for life and even after we made corrections we still need to maintain connection with the Light.