Q. When should the name of a martial art be capitalized? I see very different practices, especially for the following martial arts: Brazilian jiu-jitsu, Kodokan judo, muay Thai, and Shotokan karate. Even more confusing, some martial arts (e.g., Gracie jiu-jitsu) are both brand names and martial arts. Help!!
A. The names of martial arts that have become well known in English can usually be lowercased except for any proper noun or adjective in the name. That’s what you’ve done in your list, except that we’d write Muay Thai (with a capital M). Either because the adjective Thai follows the name for the technique (muay, roughly analogous to boxing)—or, more likely, because muay isn’t a common word in English—nobody seems to write muay Thai (with a lowercase m).
As for brand names, you could model your usage on Scotch tape and keep the generic part lowercase, as in Gracie jiu-jitsu (see CMOS 8.154). But be prepared to make exceptions for official names of organizations and the like, as in the International Brazilian Jiu-Jitsu Federation. Note that we’ve retained jiu-jitsu over the common spelling jujitsu partly based on the spelling in the name of that organization.
In sum: Use lowercase for generic words that are common in English and be prepared to make exceptions.