Lamu

In the real world Liquid soap anything was kinda expensive and dangerous until recently. But in TES we got tons of alchemists so I assume it wouldn't be for the typical wizard/mage, but it would for a commoner.

LIQUID SOAP INSTRUCTIONS;

Olive Oil

Coconut oil

crushed Coda flowers (if you wanna bathe at night, cus it glows)

crushed Comberries (if you want to make it pink and/or smell nice. other things can work too)

Wood ash/Ash, bonemeal, water (to make lye)

Water (separate from lye)

Once you made the lye and mixed the berries and coda flowers together into a paste, put the olive oil and coconut oil into a big pot and get it really hot

In a seperate bowl, add a bunch of water and the lye mixture together and mix it. It will bubble like crazy, mix it until you dont see the bubbles anymore and clear. Water first, then Lye mixture. NOT THE OTHER WAY AROUND!!

Once it's clear, put all the lye mixture and the berry-flower paste in the oil pot and give it a good stir for like 5 hours. You are allowed to take breaks but dont leave it alone for more than an hour. Maybe you can do it with magic to make it go faster, IDK. You'll start to notice at the end that its becoming more chunkier and pastey though. You wanna keep going unil it's really thick like glue.

Now, you wanna put out the fire a little and essentially slow cook the thing with a lid on it. Leave it alone for 3 hours. NOT EVEN A PEEK! This is the biggest mistake that beginner alchemists tend to make when making their first batch of liquid soap. Quit peaking at it, ya perv!

After 3 hours, it will be gel-like and somewhat transluscent. Don't put out the fire, it's not done, you gotta fold it like cake batter with a really sturdy wooden spoon so it all evens out. Once it's clear or as close to clear as possible and melts back down into a full liquid, its time to move on;

Put it all in a bunch of mason jars and wait however long it takes to become a gel-substance again.

Bring out the pot again and while it's warm (not hot) put roughly 2,365 ML of clean water (10 cups) into the pot as well as your gel-substance. Let it get hot and dissolve overnight or all day.

(optional) strain the soap of any small chunks that didn't melt before putting it in jars once again. AND BAM, YOU ARE DONE!!! You now have Liquid Soap made on Nirn.

LIQUID CONDITIONER INSTRUCTIONS;

Plant oil from leaves (NOT olive or coconut). Can be made from Bittergreen, Gold Kanet, Green Litchen, Kresh Weed, any kind of moist leaf really (do NOT use leaves found in drier regions like chokeweed), but Green Litchen or any leaves with inherent magical properties are preferred.

Clean Water

Milk, melted dreugh wax (as an emulsifier. May not seem like one logically, but it becomes one when put together)

Comberries, coda flowers (optional, it makes it smell good and glow in the dark if you wanna bathe at night) mixed into a paste

The whites of a kwamma egg or liquified/melted Muck mixed hot with sugar (so it melts). Acts like a glycerol. Make sure either of those liquids are completely clear and strained of any bits.

Plant oil (separate from the first ingredient), strained liquid from marshmerrow or any kind of clear flower tea. Acts as a liquid preservative.

Get a huge pot and put a large bowl in it. First, add the water to the pot. Put the plant oil and all the other ingredients in the large bowl. Let it heat up and melt. Strain the chunks, and mix the oiled concoction and the water together while its hot in the pot. It should all come out very thick and creamy, and also very colorful if you decide to use the berry-flower paste.

Transfer your hair mask to jars, let it completely cool to room temperature before closing the jars!

If you want it to last, though, you have to learn some simple enchanting (even a lesser soul gem could make it last a couple of years!). Otherwise, both of these would last around a week, and you're definitely not using all of it within that timeframe, even when bathing everyday.