We consider the Besov space $B^{p,q}_\alpha(G)$ on a unimodular Lie group G equipped with a sublaplacian $\Delta$. Using estimates of the heat kernel associated with $\Delta$, we give several characterizations of Besov spaces, and show an algebra property for $B^{p,q}_\alpha(G)$ for $\alpha>0$, $1\leq p,q \leq +\infty$. These results hold for polynomial as well as for exponential volume growth of balls.