On density of polynomials in the algebra of holomorphic functions of exponential type on a linear Lie group

It is shown by the author in [J. Lie Theory 29:4, 1045-1070, 2019] that for every connected linear complex Lie group the algebra of polynomials (regular functions) is dense in the algebra of holomorphic functions of exponential type. However, the argument is quite involved. Here we present a short proof.