Free lattice in three generators

Question

By general results for every set $X$ there is a free bounded lattice $L(X)$ on $X$. I would like to understand the element structure of this lattice. The cases $X=\emptyset$, $X=\{x\}$ and $X=\{x,y\}$ are quite easy. But for $X=\{x,y,z\}$ we get an infinite lattice. But what are the elements explicitly? Is there any normal form available? Or, is there any natural representation of $L(X)$? Compare this to the free group on two generators, which might be quite abstract, but it can be explicitly realized as a certain subgroup of $\mathrm{SL}(2,\mathbb{Z})$, generated by two matrices (see Ping-pong lemma). So is there a natural and non-abstract example of a bounded lattice which contains the free bounded lattice on three generators? Notice that the recursive description at Wikipedia doesn't answer these questions.