Maximal noncompact forms in classical Lie algebra? In this short note on Lie algebra, discussing about classical Lie algebra A,B,C,D class, in page 4 after Eq.(7), on the part of B,D class of O(2n,F) and O(2n+1,F) group (or algebra?), there is a statement:
"The above description given for the orthogonal algebras actually correspond to the maximal noncompact forms)."
Can someone explain what does maximal noncompact forms mean in the context of Lie algebra?
How does it relate to Eq.(7)? (ps. We know O(2n) and O(2n+1) are compact Lie groups?)
What is maximal here and what is non-compact here for the real forms? Many thanks really.
 A: From the way it is written, I suspect that "maximal noncompact form" is just the author's way to call what in most other sources is called the "split form". That sort of makes sense because in various ways, "split" vs. "compact" forms are opposite extremes. Cf. https://en.wikipedia.org/wiki/Real_form_(Lie_theory).
See how eq. 7 gives the split form, which he calls "$o(2\ell,F)$", both for $F=\Bbb{C}$ or $\mathbb{R}$. Whereas what in the next line he calls "the compact form", $o(\ell)$, is a compact Lie algebra defined only over $\Bbb{R}$. So $o(\ell)$ and $o(2\ell, \Bbb{R})$ are real Lie algebras, and are both forms of the complex Lie algebra $o(2\ell, \Bbb{C})$: the first is the compact and the second is the split form. (Over $\Bbb{C}$, there is only the split form because $\Bbb{C}$ is algebraically closed.) He plays a similar game with forms of $sp(2\ell, \Bbb{C})$ at the end of page 5 where he says "The algebras $usp(2\ell)$ and $sp(2\ell, \Bbb{R})$ are two real forms of $sp(2\ell, \Bbb{C})$. Actually $usp(2\ell)$ is a compact real form while $sp(2\ell, \Bbb{R})$ is a non-compact real form." I would expect he would say that $sp(2\ell, \Bbb{R})$ is "the maximal noncompact form" here.
If my speculation is correct, I would add that the nomenclature "maximal noncompact" is a bit unfortunate because, well, you asked the question. One might argue with something like "the $\mathfrak{p}$ in the Cartan decomposition has maximal dimension", but to just call it "the split form" seems much better.
