  We continue the investigation of the behaviour of octahedral norms in tensor
products of Banach spaces. Firstly, we will prove the existence of a Banach
space $Y$ such that the injective tensor products
$l_1\widehat{\otimes}_\varepsilon Y$ and $L_1\widehat{\otimes}_\varepsilon Y$
both fail to have an octahedral norm, which solves two open problems from the
literature. Secondly, we will show that in the presence of the metric
approximation property octahedrality is preserved from a non-reflexive
$L$-embedded Banach space taking projective tensor products with an arbitrary
Banach space.
