  In contrast with the separable case, we prove that the existence of almost
$L$-orthogonal vectors in a nonseparable Banach space $X$ (octahedrality) does
not imply the existence of nonzero vectors in $X^{**}$ being $L$-orthogonal to
$X$, which shows that the answer to an environment question in [9] is negative.
Furthermore, we prove that the abundance of almost $L$-orthogonal vectors in a
Banach space $X$ (almost Daugavet property) whose density character is
$\omega_1$ implies the abundance of nonzero vectors in $X^{**}$ being
$L$-orthogonal to $X$. In fact, we get that a Banach space $X$ whose density
character is $\omega_1$ verifies the Daugavet property if, and only if, the set
of vectors in $X^{**}$ being $L$-orthogonal to $X$ is weak-star dense in
$X^{**}$. We also prove that, under CH, the previous characterisation is false
for Banach spaces with larger density character. Finally, some consequences on
Daugavet property in the setting of $L$-embedded spaces are obtained.
