 Third-order conditions for an explicit Runge-Kutta method

For an explicit Runge-Kutta method with coefficients $A,b$, put $c=A1$ and let $c^2$ mean componentwise squaring. The conditions $b^{T}1=1$, $b^{T}c=1/2$, $b^T{c}^2=1/3$ and $b^{T}Ac=1/6$ match the exact flow expansion through degree three: the two cubic differentials are $f^{\prime \prime }[f,f]$ and $(f^{\prime }{)}^{2}f$. For sufficiently smooth vector fields these give local error $O(h^4)$ and, under the usual finite-interval Lipschitz continuity assumptions, global error $O(h^3)$.