Best conditioner/cleaner for a lambskin leather jacket?

I have a leather jacket that needs some maintenance. It’s made from lambskin, so it’s not like a hard polished cow hide. There’s a few spots I’d like to clean and I feel the leather as a whole could use a gentle conditioning.

This is the jacket

And as a bonus question: What do you do when you need to clean the inside liner of a jacket (arms/cotton interior)?