Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $M\in \mathcal{B}(F)^+$ (i.e. $M^*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in F$.

I want to show that $\mathcal{B}_1(F)$ is not a subalgebra of $\mathcal{B}(F)$, where $$\mathcal{B}_1(F)=\left\{S\in \mathcal{B}(F):\,\,\exists c>0 ;\;\langle MSy\;, \;Sy\rangle \leq c \langle My\;,\;y\rangle,\;\forall y \in \overline{\text{Im}(M)}\right\}.$$

Thank you.