i should mention that the organism can produce "alpha lipoic acid" if it can add chain length to short chain fatty acids - biotin is crucial for this matter and not a few people have biotin deficiency;
octanoic acid + cysteine (sulphur donor) -> ALA.
vitamin E and vitamin C can be recycled by the organism through ALA.
if you want to supplement with cysteine... take acetylcysteine
but you should rather/also look for cysteine (vulnerable to heat) rich foods... like oat bran. and you should keep the MSM which is in raw foods. methylsulfonylmethane gets evaporated when food is cooked at high temperature.
sulfur is key in detoxification. probiotics such as L. casei add sulfur to serine to produce further cysteine. you want that. taurine can be produced from cysteine and it is better than glycine for the purpose of conjugating things that need to be excreted by being thrown back in the gut so that it is not reabsorbed. vegetarians are usually taurine deficient because they usually choose to be vegetarians somehow irrationally, on not the right motives. no care to find out more.
https://www.ncbi.nlm.nih.gov/pubmed/7649494
and by the way, ALA is the best supplement to take before a workout due to hormesis.