Unramified Euler sums and Hoffman $\star$ basis

When looking at how periods of $\pi_{1}^{\mathfrak{m}}($\mathbb{P}$^{1}\diagdown \lbrace 0, 1, \infty \rbrace )$,i\mathrm{.}e\mathrm{.}multiplezetavalues,embedsintoperiodsof$\pi_{1}^{\mathfrak{m}}($\mathbb{P}$^{1}\diagdown \lbrace 0, \pm 1, \infty \rbrace )$,i\mathrm{.}e\mathrm{.}Eulersums,anexplicitcriteriaviathecoaction$\Delta$ acting on their motivic versions comes out. In this paper, adopting this Galois descent approach, we present a new basis for the space ${\mathcal{H}}^1$ of motivic multiple zeta values via motivic Euler sums. Up to an analytic conjecture, we also prove that the motivic Hoffman star basis $\zeta^{\star, \mathfrak{m}} (2^{a_{1}},3,\cdots,3, 2^{a_{p}}, 3, 2^{b})$isabasisof$$\mathcal{H}$^{1}$. Under a general motivic identity that we conjecture, these bases are identical. Other examples of unramified ES with alternating patterns of even and odds are also provided.