= Solution
Apply the definition to a coordinate-basis vector, use $[\partial_c,\partial_d]=0$, and collect the coefficient of $X^b$. This gives
$$
R^a{}_{bcd}
=\partial_c\Gamma^a{}_{db}-\partial_d\Gamma^a{}_{cb}
+\Gamma^a{}_{ce}\Gamma^e{}_{db}
-\Gamma^a{}_{de}\Gamma^e{}_{cb},
$$
up to the overall sign fixed in part i. The right side therefore transforms as a tensor even though its individual Christoffel-symbol terms do not. This identifies the displayed coordinate expression $K^a{}_{bcd}$ with $R^a{}_{bcd}$ after matching the paper's index and sign conventions.
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