= Solution
The <chiral spinor curvature operator> acts algebraically on a two-dimensional spinor space. We may therefore lower its matrix index using the <symplectic form> and write $\Delta_{AB}\alpha_C=X_{ABDC}\alpha^D$. Symmetry of $\Delta_{AB}$ gives symmetry in the first pair of $X$. Preservation of $\epsilon_{CD}$ says that its action is symplectic; lowering the endomorphism index of a symplectic infinitesimal transformation makes a symmetric two-index tensor. Thus $X_{ABCD}=X_{(AB)(CD)}$.
The additional <Riemann curvature tensor> symmetry $R_{abcd}=R_{cdab}$ implies $X_{ABCD}=X_{CDAB}$: project both antisymmetric spacetime index pairs onto the unprimed chiral summand. Equivalently this is the symmetry of the unprimed diagonal block of the curvature operator on <differential two-forms>. The <contracted chiral curvature identity> is another way of expressing the same three constraints on the initially nine-component pair-symmetric spinor.
Consequently $X$ is a <symmetric bilinear form> on the three-dimensional space $\operatorname{Sym}^2 S$, with six independent components. Its total symmetrization $\Psi_{ABCD}=X_{(ABCD)}$ has five components. The one-dimensional remainder is spanned by
$$
K_{ABCD}=-\epsilon_{D(A}\epsilon_{B)C}=-\frac12(\epsilon_{DA}\epsilon_{BC}+\epsilon_{DB}\epsilon_{AC}).
$$
It is symmetric within and between pairs, and $K_{(ABCD)}=0$. The latter follows from the two-dimensional alternating-form identity, or directly by evaluating its components with $\epsilon_{01}=1$. Hence the <chiral Riemann curvature spinor decomposition> is
$$
\boxed{X_{ABCD}=\Psi_{ABCD}-\Lambda\epsilon_{D(A}\epsilon_{B)C},\qquad\Psi_{ABCD}=\Psi_{(ABCD)}.}
$$
A contraction of a totally symmetric spinor with an <symplectic form> vanishes. Meanwhile the explicit two-term expression for $K$ gives $K_{ABCD}\epsilon^{AC}\epsilon^{BD}=3$. Therefore the scalar in the printed normalization is
$$
\boxed{\Lambda=\frac13X_{ABCD}\epsilon^{AC}\epsilon^{BD}.}
$$
In particular, $\Psi=X-\Lambda K$ has zero such trace and is the <Weyl curvature spinor>. The coefficient is fixed by the factor implicit in the printed symmetrization; substituting a differently normalized scalar-curvature term would change the relation between its scalar parameter and this contraction.
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