Why does $PSL(2,\mathbb C)\cong PGL(2,\mathbb C)$ but $PSL(2,\mathbb R) \not\cong PGL(2,\mathbb R)$? Why does $PSL(2,\mathbb C)\cong PGL(2,\mathbb C)$ but $PSL(2,\mathbb R) \not\cong PGL(2,\mathbb R)$?

Hint: Let $K$ be any field. $PSL_{n}(K) \simeq PGL_{n}(K)$ iff $SL_{n}(K)$ intersects all cosets of $Z(GL_{n}(K))$ in $GL_{n}(K)$
