Applicability of the Zharov–Zolotarev–Onishchik claim

Determine whether the Zharov–Zolotarev–Onishchik claim characterizing when the stabilizer of a subspace is a Lie superalgebra applies to the actions of \mathfrak{c}(\mathfrak{g}(2)) and \mathfrak{c}(\mathfrak{so}(7)) on \pi_1 and \spin_7, respectively.

Background

The classification uses stabilizers of subspaces in negative-degree components to construct many parabolic-like maximal graded subalgebras. A claim cited from Zharov, Zolotarev, and Onishchik concerns criteria under which such stabilizers are themselves Lie superalgebras.

The authors specifically leave open whether that claim applies to the central extensions of \mathfrak{g}(2) and \mathfrak{so}(7) acting on their 7-dimensional module \pi_1 and 8-dimensional spinor module \spin_7.

References

Does the Claim of , describing when the stabilizer of a~subspace $V\subset W$, not necessarily a~subsuperspace, is a~Lie superalgebra, apply to $\fg_0=\fc(\fg(2))$ and $\fc(\fo(7))$ acting on $\pi_1$ and on~$\spin_7$?

— Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras  (2610.03354 - Leites et al., 2 Oct 2026) in Section 8, Question \ref{QZZO}, “Does the Zharov--Zolotarev--Onishchik Claim apply here?”