  The tetrahedron algebra $\boxtimes$ is an infinite-dimensional Lie algebra
defined by generators $\{x_{ij} \mid i, j \in \{0, 1, 2, 3\}, i \neq j\}$ and
some relations, including the Dolan-Grady relations. These twelve generators
are called standard. We introduce a type of element in $\boxtimes$ that "looks
like" a standard generator. For mutually distinct $h, i, j, k \in \{0, 1, 2,
3\}$, consider the standard generator $x_{ij}$ of $\boxtimes$. An element $\xi
\in \boxtimes$ is called $x_{ij}$-like whenever both (i) $\xi$ commutes with
$x_{ij}$; (ii) $\xi$ and $x_{hk}$ satisfy a Dolan-Grady relation. Pick mutually
distinct $i,j,k \in \{0,1,2,3\}$. In our main result, we find an attractive
basis for $\boxtimes$ with the property that every basis element is either
$x_{ij}$-like or $x_{jk}$-like or $x_{ki}$-like. We discuss this basis from
multiple points of view.
