Do adjoint functors really define monads? It is often claimed as "obvious" that a pair of adjoint functors: $L\colon{\cal V}\to {\cal M}$ and $R\colon{\cal M}\to {\cal V}$ defines a cotriple $(\bot, \epsilon, \delta)$ and a monad. What is wrong with the following counterexample?
Let ${\cal V}$ be the category of vector spaces over $\mathbb R$ and ${\cal M}$ be the category of modules over a Lie algebra $\mathfrak g.$ Then (I believe) the forgetful functor $R:{\cal M}\to {\cal V}$ is right adjoint to $L\colon{\cal V}\to {\cal M},$ $L(V)=V\otimes \mathfrak g.$ 
The counit $\epsilon\colon M\otimes \mathfrak g\to M$ is given by the action of $\mathfrak g$ on $M.$ In Weibel's "Intro to Homological Algebra", the construction of a monad is based on the following identity: $\epsilon\circ (LR\epsilon)=\epsilon\circ(\epsilon LR)$. That means that $(m\cdot x)\cdot y=m\cdot (x\cdot y)$ for $x,y\in \mathfrak g.$ But for Lie algebra modules we have $(m\cdot x)\cdot y=m\cdot (x\cdot y)+(m\cdot y)\cdot x$!

I think I found the error: My $L$ and $R$ are not adjoint! For an associative algebra $A,$ over $\mathbb R$, the forgetful functor $R$ is right adjoint to $L: {\mathcal V}\to {\mathcal M}$ (= category of right $A$-modules) by sending for example $z\in M=Hom_{\mathbb R}({\mathbb R},M)$ to $r_z\in Hom_{A}(A,M),$ $r_z(a)=z\cdot a.$ This is an $A$-module homomorphisms because $r_z(ab)=r_z(a)b$. But that fails when $A$ is not associative!
Thank you all for your comments, especially @Omar who pointed in that direction!
