Show that a metrizable Abelian topological group can be metrized by an invariant metric $d$. My efforts:
Let the Abelian topological group $G$ be metrized by $\rho$, which is not invariant. We want to construct an invariant metric $d$ from $\rho$.
Define $\widetilde{\rho}(x,y):=$ max{$\rho(x,y),1$}. Then define $d(x,y):=$ sup$_{z\in G}|\widetilde{\rho}(z+x,0)-\widetilde{\rho}(z+y,0)|$. The sup must exist since $\widetilde{\rho}$ is continuous and bounded between 0 and 1.
We show $d$ is an invariant metric. Obviously $d(x,y)\geq 0$, $d(x,y)=d(y,x)$, $d(x,x)=0$, and $d(x+z,y+z)=d(x,y)$.
Next we show $d(x,y)=0\implies x=y$. If $d(x,y)=0$, i.e., the sup of a nonnegative function $|\widetilde{\rho}(z+x,0)-\widetilde{\rho}(z+y,0)|$ of $z$ is zero, it must be zero for all $z\in G$ and this can only happen when $x=y$.
The remaining thing to show is the triangular inequality.
$d(x,y)=$ sup$_{z\in G}|\widetilde{\rho}(z+x,0)-\widetilde{\rho}(z+y,0)|\leq$ sup$_{z\in G}(|\widetilde{\rho}(z+x,0)-\widetilde{\rho}(z+w,0)|+|\widetilde{\rho}(z+w,0)-\widetilde{\rho}(z+y,0)|)$
$\leq$ sup$_{z\in G}|\widetilde{\rho}(z+x,0)-\widetilde{\rho}(z+w,0)|$ + sup$_{z\in G}|\widetilde{\rho}(z+w,0)-\widetilde{\rho}(z+y,0)|=d(x,w)+d(w,y)$.
We only need to show that $d$ induces the same topology as $\widetilde{\rho}$, since $\widetilde{\rho}$ is also a metric inducing the same topology as $\rho$.
Let $x\in G$ and $\epsilon>0$ be given. We need to find a $\delta>0$ such that $B_d(x,\delta)\subset B_\widetilde{\rho}(x,\epsilon)$, i.e., sup$_{z\in G}|\widetilde{\rho}(z+x,0)-\widetilde{\rho}(z+y,0)|<\delta\implies\widetilde{\rho}(x,y)<\epsilon$.
I don't know how to find $\delta$.
 A: The required claim is a corollary of well-known theorem (see, for instance, [AT, Corollary 3.3.13]) stating that a $T_0$ topological group is metrizable by a left-invariant metric iff it is first-countable. The proof of the theorem is well-known but rather long and technical. If you need details, I can copy for you the respective pages (151-155) from [AT].
On the other hand, I started the list of my publications from the paper [R] on constructing metrics on topological groups and I don’t see a direct construction of an invariant metric $d$ from the given metric $\rho$. Although, of course, I can miss some simple way, I guess we can encounter problems in straightforward attempts. For instance, if in your example we take $G=\Bbb R$ with $\rho(x,y)=|x^3-y^3|$, for each $x,y\in\Bbb R$ then we obtain $d(x,y)=1$ for each distinct $x,y\in\Bbb R$.
References
[AT] Alexander Arhangel'skii, Mikhail Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.
[R] Alex Ravsky A. On extension of (pseudo-)metrics from subgroup of topological group onto the group,  Matematychni Studii {\bf 11}:1 (1999) 31–39. (Example 2 is wrong and so Remark 7 should be modified too).
