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.