Maximality of the four max(b) subalgebras

Prove the maximality of the four exceptional graded subalgebras whose occurrences are marked max(b): \mathfrak{sl}(2)\oplus\mathfrak{sl}(2|1) and \mathfrak{osp}(1|2)\oplus\mathfrak{osp}(1|2) in \mathfrak{ag}(2), and \mathfrak{g}(2)\oplus\mathfrak{osp}(1|2) and \mathfrak{osp}(1|2)\oplus\mathfrak{osp}(1|2)\oplus\mathfrak{sl}(2) in \mathfrak{ab}(3).

Background

The paper’s maximality criterion is conclusive when the relevant torus acts with one-dimensional weight spaces. Four listed subalgebras are marked max(b) because their intersection with the chosen maximal torus is too small for that criterion to rule out intermediate subalgebras.

The authors report that computational tests found no intermediate subalgebras, but they do not regard this as a proof. A rigorous maximality proof is therefore left unresolved for these four cases.

References

Prove the maximality of the four answers all of whose occurrences are marked max(b), i.e., of $\fsl(2)\oplus\fsl(2|1)$ and $\fosp(1|2)\oplus\fosp(1|2)$ in $\fag(2)$, and of $\fg(2)\oplus\fosp(1|2)$ and $\fosp(1|2)\oplus\fosp(1|2)\oplus\fsl(2)$ in $\fab(3)$; for them the torus $\ft'=\fu\cap\ft$ is too small for Lemma $\ref{Lcert3exc}$ to be conclusive.

— Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras  (2610.03354 - Leites et al., 2 Oct 2026) in Section 8, Problem \ref{Pmaxb}, “Maximality of the four answers marked max(b)”

Can the answers be stated in terms of odd reflections? Each of the gradings of tables tabOspGr--tabAbGr is given by an element $h$ of a~maximal toral subalgebra, i.e., by a~subset of a~system of simple roots; the systems of simple roots of the three exceptions form a~groupoid under odd reflections, see , and the answers should be equivariant with respect to it.

— Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras  (2610.03354 - Leites et al., 2 Oct 2026) in Section 8, Question \ref{Qoddrefl}, “The answers in terms of odd reflections”