Coupled angular-momentum state count ($(2J_{+}+1)+(2J_{-}+1)=4L+2$)

Coupling orbital angular momentum $L>0$ to spin one-half produces $J=L\pm 1/2$. The dimensions add as $(2L+2)+2L=2(2L+1)$. A change between coupled and product bases changes labels, not the dimension of the Hilbert space.