The tetrahedron algebra ⊠ is an infinite-dimensional Lie algebra defined by generators {xij|i,j∈{0,1,2,3},i≠j} and some relations, including the Dolan-Grady relations. These twelve generators are called standard. We introduce a type of element in ⊠ that “looks like” a standard generator. For mutually distinct h,i,j,k∈{0,1,2,3}, consider the standard generator xij of ⊠. An element ξ∈⊠ is called xij-like whenever both (i) ξ commutes with xij; (ii) ξ and xhk satisfy a Dolan-Grady relation. Pick mutually distinct i,j,k∈{0,1,2,3}. In our main result, we find an attractive basis for ⊠ with the property that every basis element is either xij-like or xjk-like or xki-like. We discuss this basis from multiple points of view.