I am familiar with a few of them.
Potassium palmate - this is the soap (potassium salt) formed from potassium hydroxide (KOH) and palm oil.
Sodium tallowate, - sodium salt - soap formed from tallow (beef fat) and sodium hydroxide NaOH
Sodium palmate, - soap obtained from NaOH and palm oil.
Sodium palm kernelate, - soap formed from NaOH and palm kernel oil.
Glycerin, - a humectant which is a natural product from soap reaction (or it can be added in, usually is for a shave soap, for longer lived lather)
Parfum, - fragrance
Sodium chloride, - just sea salt or table salt
Petrolatum, - petroleum jelly
Palm kernel acid, ??? (do they mean fatty acids)
Orthotolyl biguanide, ???
Linalool, ???
Limonene, - this is a component in some citrus essential oils
Farnesol, ???
Citral, - another component in citrus essential oils.
Tetrasodium etidromate, ???
Pentasodium, ???
pentetate, ???
Tetrasodium EDTA, - this is a chemical preservative, it is used to slow down rancidity in soaps.
Cl 77891 ????
I left a blank for the ones i don't know. Most of them are probably preservatives of some sort, the Palm kernel acid might be extra Palm kernel oil for superfatting? I am sure other folks here know the rest of the chemicals. Even if not, you can figure out this soap.
So basically both types of lye - KOH and NaOH - were used to make this soap. KOH was only reacted with Palm oil, (the most abundant ingredient) and NaOH was used to saponify tallow, palm oil, and palm kernel oil. Extra glycerin was probably added (add to the lye water) and some citrus was used to scent the soap. Oh yes, and salt was used in the soap too, probably to harden it due to using KOH for part of the soap (KOH only makes soft or paste soap so salt is used to harden it).
There are a few gaps but I hope this helps some.