= Solution
The criterion for <algebraic stability of a Runge-Kutta method> is $b_i\geq0$ and positive semidefiniteness of
$$
M=BA+A^TB-bb^T,\qquad B=\operatorname{diag}(b).
$$
The <Runge-Kutta contractivity identity> implies that this criterion is sufficient for <B-stability>, namely contractivity for a <dissipative vector field>, whenever the stage equations are defined. The implication from <algebraic stability> to <B-stability> is the relevant nonlinear theorem; <A-stability> alone is a linear property.
All weights here are positive, but
$$
M=\frac1{36}\begin{pmatrix}-1&1&0\\1&0&-1\\0&-1&1\end{pmatrix},
\qquad e_1^TMe_1=-\frac1{36}<0.
$$
A <positive semidefinite matrix> cannot have this negative quadratic value. Hence \b[the method is not <algebraically stable>]. Failure of this sufficient criterion alone would not be a proof of failure of <B-stability>; no such converse is needed for the question.
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