Uniform doubling for abelian products with $\mathrm{SU}(2)$

We prove that the uniform doubling property holds for every Lie group which can be written as a quotient group of $\mathrm{SU}(2)\times {\mathbb{R}}^n$ for some $n$. In particular, this class includes the four-dimensional unitary group $\mathrm{U}(2)$. As this class contain non-compact as well as compact Lie groups, we discuss a number of analytic and spectral consequences for the corresponding heat kernels.