The Chevalley Group F4(Z) Is Generated by Three Involutions, Two of Which Commute

In Problem 15.67 of the Kourovka Notebook (2002), Ya. N. Nuzhin asked which adjoint Chevalley groups of normal type over the integers are generated by three involutions, two of which commute. By work of Tamburini and Zucca, Nuzhin, Timofeenko, and Kazakova and Nuzhin, the answer was known for the types \(A_l\), \(B_2=C_2\), \(G_2\), \(E_6\), \(E_7\) and \(E_8\); the types \(B_l\), \(C_l\) (\(l\ge3\)), \(D_l\) (\(l\ge4\)) and \(F_4\) were open. Our result is a partial answer: the adjoint Chevalley group \(F_4(\mathbb{Z})\) is generated by three involutions two of which commute (computer-assisted proof with an exact certificate); the types \(B_l\), \(C_l\) (\(l \ge 3\)) and \(D_l\) (\(l \ge 4\)) stay open. The involutions are explicit and follow the pattern of Timofeenko's triples for \(E_l\): \(b\) and \(c\) lift the two halves of the bipartite Coxeter element, and \(a\) is a product of commuting root elements and a torus element (four root elements of both lengths here, two in Timofeenko's triples). A direct argument shows that they are involutions and reduces \(ab=ba\) to four signs of structure constants. Generation is proved by a straight-line program with \(1064\) nodes which produces the \(48\) root elements \(x_r(1)\); a checker that uses only the Python standard library verifies it exactly in a few seconds, and three independent programs confirm it. The formulas refer to a Chevalley basis that we describe by its extraspecial signs; the existence statement does not depend on this choice. As consequences, the reductions modulo every prime \(p\) generate \(F_4(p)\) in the same way, and the least number of generating involutions of \(F_4(\mathbb{Z})\) with product \(1\) is \(5\). No printed answer for \(F_4\) was found in the sources searched (zbMATH, arXiv, OpenAlex, Crossref, Semantic Scholar, web searches); a search that finds nothing is not a proof of novelty. This is an unrefereed note.