The dimensionality of the indefinite orthogonal group $O(p, q)$ equals $n(n-1)/2$, where $n := p + q$. Is it similarly true that the dimensionality of all the indefinite Lie groups with signature $(p, q)$ depends only on $n := p + q$? For example, is the dimensionality of $U(p, q)$ equal to $(p + q)^2$?
This statement seems intuitively obvious, because we should be able to convert any $U(p, q)$ matrix to a $U(p + q)$ matrix simply by judiciously inserting some factors of $i$, but I cannot prove it.