  Let $R$ be an algebra essentially of finite type over a field $k$ and let
$\Omega_k(R)$ be its module of K\"ahler differentials over $k$. If $R$ is a
homogeneous complete intersection and $\mathrm{char}(k)=0$, we prove that
$\Omega_k(R)$ is of linear type whenever its Rees algebra is Cohen-Macaulay and
locally at every homogeneous prime $\mathfrak{p}$ the embedding dimension of
$R_{\mathfrak{p}}$ is at most twice its dimension.
