How could I use "respectively" to state three actions by subsets of four people?

If your aim is accuracy, I think you may need two independent clauses since the teams of Arnold + Benson and Arnold + Davidson proved two claims but not three while Carlson proved only one. How about:

When X is a rainbow set or a rainy set, the claim was proved by Arnold and Benson and Arnold and Davidson, respectively; Carlson further proved the claim when X is a cloudy set.

This construction removes the ambiguity inherent with your version. The word respectively makes more sense when the number of researchers matches the number of claims or proofs, as in my first clause.

This may be a case where the ampersand is useful. It may be used to combine two or more items to a composite singular noun. For example, "Smith & Nephew" is a pharmaceutical company.

Smith & Nephew plc, also known as Smith+Nephew, is a British multinational medical equipment manufacturing company headquartered in Watford, England.

Wikipedia

David Speaker identifies the ampersand as acceptable:

inside graphic or document tables or within parentheses when space is limited; in common shorthand expressions such as “rock & roll”; within a series to identify an item as part of its name and not a separator (e.g., “rock, pop, rhythm & blues, and hip-hop”);

and in identifying more than one addressee, particularly a couple: “Mr. & Mrs. Johnson.”

Grammar Book

Your sentence is thus easily modified to put the three sets of authors into one-to-one registration with their proofs:

When X is a rainbow set, a cloudy set, and a rainy set, the claim was proved by Arnold & Benson [1], Carlson [2], and Arnold & Davison [3], respectively.