Let $\mathrm{so}(p,q)$ be the real definite/indefinite orthogonal Lie algebra, $p,q\ge0$, $p+q=n\in\mathbb{N}$, and $L\subset\mathrm{so}(p,q)$ a Lie subalgebra with non-trivial centre, $\mathrm{Z}(L)\neq0$.

**Question:**

Is there an upped bound $c(p,q)$ on the dimension of $L$, $\dim L\le c(p,q)$, better than the dimension of the maximal proper subalgebra?

**Discussion:**

I have tried to brute-force a solution through a classification of closed subgroups of classical groups as in

M. Liebeck, G. Seitz, "On the subgroup structure of classical groups", Invent. Math. 134, 1998.

But the option (ii) of Theorem 1 is not very explicit, and I cannot extract any dimension-relevant  information from there.

Thank you.