It is established that in a commutative radical Fréchet algebra, elements of non-nilpotent finite closed descent exist if a locally non-nilpotent element of locally finite closed descent exists. Thus if $\mathbb{C}[[X]]$ can be embedded into the unitization of the algebra in such a way that $X$ is mapped to an element which is locally non-nilpotent, then it is possible to embed the ‘structurally rich’ algebra $\mathbb{C}_{\omega_{1}}$.