  We identify all closed Lie ideals of $A \otimes^{\alpha} B$ and $B(H)
\otimes^{\alpha} B(H)$, where $\otimes^{\alpha}$ is either the Haagerup tensor
product, the Banach space projective tensor product or the operator space
projective tensor product, $A$ is any simple C*-algebra, $B$ is any C*-algebra
with one of them admitting no tracial states, and $H$ is an infinite
dimensional separable Hilbert space. Further, generalizing a result of Marcoux,
we also identify all closed Lie ideals of $A\otimes^{\min} B$, where $A$ is a
simple C*-algebra with at most one tracial state and $B$ is any commutative
C*-algebra.
