(In)dependence of the axioms of $\Lambda$-trees

A $\Lambda$-tree is a $\Lambda$-metric space satisfying three axioms (1), (2) and (3). We give a characterization of those ordered abelian groups $\Lambda$ for which axioms (1) and (2) imply axiom (3). As a special case, it follows that for the important class of ordered abelian groups $\Lambda$ that satisfy $\Lambda=2\Lambda$, (3) follows from (1) and (2). For some ordered abelian groups $\Lambda$, we show that axiom (2) is independent of the axioms (1) and (3) and ask whether this holds for all ordered abelian groups. Part of this work has been formalized in the proof assistant \textsf{Lean}.