Lie Superoperator Algebra Commutation relations

I am trying to derive commutation relation of Lindblad Superoperator relations.
They are defined as:
$$cdot a^{dagger}a=mathcal{L}$$
$$a^{dagger}acdot =mathcal{R}$$
$$acdot a^{dagger}=mathcal{M}$$

I try to derive this commutation relation $[mathcal{L},mathcal{M}]=-mathcal{M}$ using the definition of the superoperators:
$$(Acdot) [rho] = Arho $$
$$(cdot A) [rho] = rho A $$

From definitions above we can conclude some properties of superoperators:

$$Acdot B cdot =(Acdot) [B cdot] = ABcdot $$
$$cdot A cdot B =(cdot A) [cdot B] =cdot BA $$

Now when I try to derive this relation, it reads as
$$[mathcal{L},mathcal{M}]=[cdot a^{dagger}a,acdot a^{dagger}]=(cdot a^{dagger}aa)[cdot a^{dagger}]-a(cdot a^{dagger})[cdot a^{dagger}a]=cdot a^{dagger} a^{dagger}aa-acdot a^{dagger}aa^{dagger}
=cdot a^{dagger} a^{dagger}aa-acdot a^{dagger}a^{dagger}a-acdot a^{dagger}$$
where in the last line I used relation $[a,a^{dagger}]=1$. The thing is, I do not know how to get rid of the first two terms in the last expression. If they are got rid of, then I get the commutation relation I want. Any ideas?