6. As shown in Figure 3, in rhombus $A B C D$, $\angle A B C=$ $120^{\circ}, A B=\sqrt{3}, E$ is a point on the extension of side $B C$, $A E$ intersects $C D$ at point $F$, connect $B F$ and extend it, intersecting $D E$ at point $G$. Then the maximum length of line segment $B G$ is ( ).
(A) $\sqrt{3}$
(B) 2
(C) $2 \sqrt{3}$
(D) 4