Let $M= L \mathbb F_2$ and $H = \ell^2 \mathbb F_2$, where $\F_2 = \langle a,b \rangle$ is the free group on two generators and $B(H)$ is a bimodule via the left and right multiplication with the left-regular representation $\lambda \colon L \mathbb F_2 \to B(\ell^2 \mathbb F_2)$.

Define $\delta(x) = [x,\lambda(a)]$. Then, $\lambda(a^{\pm1})'' \subset L \mathbb F_2$ is a MASA (as can be shown) and $\delta$ vanishes on it. However, $\delta(\lambda(b)) \neq 0$ so that $\delta$ does not vanish on $L \mathbb F_2$.

Let $M= L \mathbb F_2$ and $H = \ell^2 \mathbb F_2$, where $\mathbb F_2 = \langle a,b \rangle$ is the free group on two generators and $B(H)$ is a bimodule via the left and right multiplication with the left-regular representation $\lambda \colon L \mathbb F_2 \to B(\ell^2 \mathbb F_2)$.

Define $\delta(x) = [x,\lambda(a)]$. Then, $\lambda(a^{\pm1})'' \subset L \mathbb F_2$ is a MASA (as can be shown) and $\delta$ vanishes on it. However, $\delta(\lambda(b)) \neq 0$ so that $\delta$ does not vanish on $L \mathbb F_2$.