Multiplicative type groups split over a finite Galois extension

The affineness and splitting data supplied by F1 give $A$ and the extension $K$, so the calculation below takes place in the coordinate algebra of $G$ over the splitting field. For a finite subcoalgebra $C\subset A$, the finite algebra $E=C^{\ast }$ is commutative (commutators vanish after the faithful extension to $K$) and satisfies $E\otimes K\cong K^d$ by F2. For each $a\in E$, let $f_a$ be its minimal polynomial over $k$. The independent powers preceding its degree stay independent under extension, so $f_a$ is also the minimal polynomial over $K$. In $K^d$, that polynomial is the product of the distinct linear factors associated to the coordinate values of $a$. Thus $f_a$ is separable over $k$. Choose a finite algebra generating set of $E$, for example a vector-space basis. Over $k_s$, each generator has a split squarefree minimal polynomial, whose Lagrange interpolation idempotents decompose the algebra into factors on which that generator is a scalar. Repeating with the finitely many generators decomposes $E\otimes k_s$ into factors generated only by scalars, hence copies of $k_s$. Dualizing shows that $C\otimes k_s$ is spanned by group-like elements. Every element of $A$ belongs to a finite subcoalgebra by F2, so $A\otimes k_s$ is spanned by its group-like elements.

Distinct group-like elements in any coalgebra are linearly independent. Otherwise take a shortest relation, write one as $g={\sum }_{i=1}^r{a}_i{g}_i$ with the $g_i$ independent and all $a_i\mathrm{\ne }0$, and compare $\mathrm{\Delta }(g)$ with $g\otimes g$. In the independent tensor family $g_i\otimes g_j$, the off-diagonal coefficients give $a_i{a}_j=0$ for $i\mathrm{\ne }j$, so $r=1$; the diagonal and counit then give $a_1=1$, contrary to distinctness. In a Hopf algebra the group-like elements form an abelian group under multiplication with inverse $S$. Therefore their spanning and independence identify $A\otimes k_s$ with $k_s[M]$ as a Hopf algebra. F3 implies that $M$ is finitely generated.

Choose finite generators $m_1,\dots ,m_r$ of $M$. Their corresponding group-like elements are finite sums of tensors, so all their coefficients lie in a finite separable extension of $k$. Enlarge it inside $k_s$ by adjoining all roots of the finitely many separable minimal polynomials of its generators. This gives a finite normal separable extension $L$, which is Galois by F4. The corresponding group-like elements and their inverses define a Hopf map $L[M]\to A\otimes L$ which becomes the isomorphism of step 2.1 after extension to $k_s$. A map of vector spaces is injective and surjective if it becomes so after field extension: kernels and cokernels tensor exactly, and a nonzero vector stays nonzero. Thus the map is already an isomorphism over $L$.