We give sufficient conditions for hypoellipticity of a second order operator with real-valued infinitely differentiable coefficients whose principal part is the product of a real-valued infinitely differentiable function φ(
x) and the sum of squares of first order operators
X1,…,
Xr. These conditions are related to the way in which φ(
x) changes its sign, and the rank of the Lie algebra generated by φ
X1,…,φ
Xr and
X0 where
X0 is the first order term of the operator. Our result is an extension of that of [
4], and it includes some cases not treated in [
1], [
5] and [
8].
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