  Denote by $\epsilon$ a primitive root of $N^{th}$-unity. In this paper, we
show that the unit cyclotomic multiple zeta values for $\mu_N$ generate all the
cyclotomic multiple zeta values for $\mu_N$ in cases $N=2,3,4$. Moreover, the
unit cyclotomic multiple zeta values for $\mu_N$ can be written as
$\mathbb{Q}$-linear combinations of $\left(\zeta\binom{1}{\epsilon}\right)^n,
\left(\zeta\binom{1}{\epsilon^{-1}}\right)^n$ and lower depth terms in each
weight $n$ in case of $N=2,3$ and $4$. By detailed analysis of the motivic
Galois action, we compute the coefficients of
$\left(\zeta\binom{1}{\epsilon}\right)^n,
\left(\zeta\binom{1}{\epsilon^{-1}}\right)^n$ in the above expressions of unit
cyclotomic multiple zeta values.
