= Solution
The coefficients are those of the three-stage <Lobatto IIIA method>. Set $e=(1,1,1)^T$, $c=Ae=(0,1/2,1)^T$ and $C=\operatorname{diag}(c)$. Direct calculation gives all eight <fourth-order conditions for a Runge-Kutta method>:
$$
\begin{gathered}
b^Te=1,\quad b^Tc=\tfrac12,\quad b^Tc^2=\tfrac13,\quad b^TAc=\tfrac16,\\
b^Tc^3=\tfrac14,\quad b^TCAc=\tfrac18,\quad
b^TAc^2=\tfrac1{12},\quad b^TA^2c=\tfrac1{24}.
\end{gathered}
$$
Powers of $c$ are componentwise. These <Butcher order conditions> prove order at least four for a smooth general nonlinear vector field; matching a scalar test equation alone would not suffice. A necessary fifth-order condition fails:
$$
b^Tc^4=\frac5{24}\ne\frac15.
$$
Hence the method has \b[order exactly four], with one-step <local truncation error> $O(h^5)$ and, under stability and smoothness hypotheses, <global error> $O(h^4)$.
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