Physicist here. I seem to see conflicting statements about the rank of some groups I've come across lately.
A paper I'm reading states that $SO(6)$ is rank 3 and therefore its Cartan subalgebra has three commuting generators. But I also read elsewhere that $SO(N)$ has rank $N(N-1)/2$, ie, equal to its number of generators, here 15. Similarly I see it stated that $SU(4)$ is rank 3 (as it better be, under some understanding, since $SO(6) \sim SU(4)$) but again I am more familiar with its rank being defined as the number of its generators, ie, $4^2-1=15$ here, $N^2-1$ for general $SU(N)$.
I suppose these are just two different uses of the word rank? Is one usage sloppy? (edit: It sounds like one usage is not just sloppy, but actually wrong)
Perhaps the number of generators is more properly called the group's dimension. But then there is also the dimension of a representation, which is not in general the same as the number of generators for the group. Correct? Like $SU(2)$ in its fundamental representation acts on a vector space of dimension 2, but it has 3 generators.
I know in some places the rank is specifically defined as the number of generators in the Cartan subalgebra. If that is the case, how else could one determine the rank? I wonder because the author states the rank of these groups is 3 and then deduces from that the number of Cartan generators.
Sorry for any sloppy terminology or general ignorance, mathematicians!
If there's some place where all this is gathered together and made clear, I'll happily take any references you might give.