Let $N,M$ be normal subgroups of $G$ with $N\cap M=\{e\}$. Prove that $M\subset C_{G}(N)$ and $N\subset C_{G}(M)$. First consider the following definition:

Let $G$ be a group and $H$ a subgroup of $G$. The center:
$$ C_{G}(H)=\{g\in G\,:\,gh=hg,\,\forall h\in H\}$$

Now I'm trying to prove the following theorem:

Let $N,M$ be normal subgroups of $G$ so $N\cap M=\{e\}$.
Prove that $M\subset C_{G}(N)$ and $N\subset C_{G}(M)$.

What I tried:
We know that $N\triangleleft G$ so there is $n_{0}\in N$ so for every $n\in N$ and every $g\in G$, $n_{0}=g^{-1}ng$. We can convert it to be: $n=gn_0g^{-1}$.
We also know that $M\triangleleft G$ so there is $m_{0}\in M$ so $m=gm_0g^{-1}$.
We can see that:
$$ nm=(gn_0g^{-1})(gm_0g^{-1})=g(n_0m_0)g^{-1}\\
   mn=(gm_0g^{-1})(gn_0g^{-1})=g(m_0n_0)g^{-1}$$
But how I continue from here? As I understand I need to show that $nm=mn$.

Observe that
$$K\lhd G\iff [K,G]=1$$
So apply this:
$$M\lhd G\iff [M,G]=1\;,\;\;\text{and in particular}\;\;\forall\,m\in M\,,\,n\in N: [m,n]=1\implies$$
$$m^{-1}n^{-1}mn=1\implies mn=nm\implies M\subset C_G(N)$$
