It seems really obvious in hindsight, so I'm sure there's just prior art I don't know about.
There are, in general, a number of different sorting algorithms which are optimized for a specific number of elements. In this case, four. It uses five binary comparisons to sort four elements.
You can find other algorithms like this, such as an algorithm that uses seven comparisons to sort five items, or one that uses ten comparisons to sort six items.
In theory, you can get O(n log k) performance, where k is the number of distinct elements (so k <= n). And crucially: you don't need to know k up-front.
In practice, all my attempts were absolutely slower than the standard approach based on binary comparisons only. (But that's saying more about me than about the domain.)