Group-like elements and primitives relative to g in a two-dimensional coalgebra

The displayed assignments extend linearly by [F4]. On $g$, coassociativity and both counit identities follow from $\mathrm{\Delta }(g)=g\otimes g$ and $\epsilon (g)=1$. On $x$, the left coassociativity composite is $x\otimes g\otimes g+g\otimes x\otimes g+g\otimes g\otimes x$, and the right composite gives the same three terms; the left and right counit contractions of $\mathrm{\Delta }(x)$ each equal $x$. Since the coalgebra identities are equalities of linear maps and $g,x$ form a basis, [F1] holds.

Write $y=ag+bx$. If $y$ is group-like, the coefficient of $x\otimes x$ in $\mathrm{\Delta }(y)=y\otimes y$, compared in the basis from [F3], gives $b^2=0$. If $b\mathrm{\ne }0$, multiplication by $b^{-1}$ from [F5] gives $b=0$, a contradiction; hence $b=0$. Then $\epsilon (y)=a=1$, so $y=g$, which is indeed group-like by the displayed formulas and [F2].

If $y=ag+bx$ is primitive relative to $g$, then $\epsilon (y)=a=0$ by [F2]. Conversely, for every $b\in k$, $\mathrm{\Delta }(bx)=b(x\otimes g+g\otimes x)=bx\otimes g+g\otimes bx$ and $\epsilon (bx)=0$ by linearity; thus precisely the vectors $bx$ are $g$-primitive.