  Let G be a noncompact connected Lie group and $\rho$ be the right Haar
measure of G. Let $X_1,...,X_q$ be a family of left invariant vector fields
which satisfy H\"ormander's condition, and let $\Delta=-\sum_{i=1}^qX_i^2$ be
the corresponding subLaplacian.
  For $1\leq p<\infty$ and $\alpha\geq 0$ we define the Sobolev space
$L^p_{\alpha}(G)={f in L^p(\rho): \Delta^{\alpha/2}f\in L^p(\rho) }$, endowed
with the norm $\|f\|_{\alpha,p}=\|f\|_{p}+\|\Delta^{\alpha/2}f\|_p$, where we
denote by $\|f\|_p$ the norm of $f$ in $L^p(\rho)$.
  In this paper we show that for all $\alpha\geq 0$ and $p\in (1,\infty)$, the
space $L^{\infty}\cap L^p_{\alpha}(G)$ is an algebra under pointwise product.
Such result was proved by T. Coulhon, E. Russ and V. Tardivel-Nachef in the
case when G is unimodular. We shall prove it on Lie groups, thus extending
their result to the nonunimodular case.
  In order to prove our main result, we need to study the boundedness of local
Riesz transforms $R^c_J=X_J(cI+\Delta)^{-m/2}$, where c>0,
$X_J=X_{j_1}...X_{j_m}$ and $j_\ell \in\{1,\dots,q\}$ for $\ell=1,...,m$. We
show that if c is sufficiently large, the Riesz transform R^c_J is bounded on
$L^p(\rho)$ for every $p\in (1,\infty)$, and prove also appropriate endpoint
results involving Hardy and BMO spaces.
