$S_3$ is split over the rationals

The line $\mathbb{Q}(1,1,1)$ is trivial and its invariant complement $W$ has dimension two. With $u=(1,-1,0)$ and $v=(0,1,-1)$ as a basis of $W$, the transposition $s=(12)$ and cycle $r=(123)$ act by $S=\left(\begin{array}{cc}-1& 1\\ 0& 1\end{array}\right)$ and $R=\left(\begin{array}{cc}0& -1\\ 1& -1\end{array}\right)$. In particular $R$ has no eigenvalue $1$ over $\mathbb{C}$.

Every one-dimensional complex representation of $S_3$ is trivial or sign: conjugate transpositions have one common image $\epsilon$ with ${\epsilon }^2=1$, and every $3$-cycle is a product of two transpositions, so acts as $1$. A proper invariant line in $W_{\mathbb{C}}$ would therefore be fixed by $r$, contrary to step 1.1. Thus $W_{\mathbb{C}}$ and $W$ are irreducible.

The two central averages $e_{+}=\frac{1}{6}{\sum }_{g\in S_3}g$ and $e_{-}=\frac{1}{6}{\sum }_{g\in S_3}\mathrm{sgn}(g)g$ act respectively as $(1,0,0)$ and $(0,1,0)$ under the algebra map $\mathrm{\Phi }:\mathbb{Q}[S_3]\to \mathbb{Q}\oplus \mathbb{Q}\oplus {\mathrm{End}}_{\mathbb{Q}}(W)$ defined by the three displayed representations; they kill $W$ because $W$ has neither a trivial nor a sign line. The four matrices $I,S,R,SR$ are linearly independent over $\mathbb{Q}$ (their coordinate determinant is $-3$), so the $W$-projection of $\mathrm{\Phi }$ is all $M_2(\mathbb{Q})$. The central averages separate the scalar factors, making $\mathrm{\Phi }$ surjective; both sides have dimension $6$, hence $\mathrm{\Phi }$ is an isomorphism.

Tensoring step 3.1 with $\mathbb{C}$ gives $\mathbb{C}[S_3]\cong \mathbb{C}\oplus \mathbb{C}\oplus M_2(\mathbb{C})$. By [L1] over both fields, the three displayed models are all irreducibles over $\mathbb{Q}$ and $\mathbb{C}$. Each simple module of $\mathbb{Q}\oplus \mathbb{Q}\oplus M_2(\mathbb{Q})$ has endomorphism ring $\mathbb{Q}$: this is immediate for the scalar factors, while an endomorphism of the column module commuting with every matrix unit is scalar. The splitting-field definition therefore makes $\mathbb{Q}$ a splitting field for $S_3$.