Example 4

The complex $C$ is split as soon as $B(C)$ and $H(C)$ are projective (resp. as soon as $B_n(C)$ and $H_n(C)$ are injective for each $n$). Indeed, according to the exact sequences $(I_n)$ to $(IV_n)$ of No. 1, $Z(C)$ is then a direct factor in $C$ and $B(C)$ a direct factor of $Z(C)$ (resp. $B(C)$ is a direct factor in $C$ and $Z(C)/B(C)$ a direct factor in $C/B(C)$).