Is $L=\left \{ wxw \,\,|\,\,w,x \,\in \left ( a+b \right )^{+}\right \}$ Regular? Question

Is $L=\left \{ wxw \,\,|\,\,w,x \,\in \left ( a+b \right )^{+}\right \}$ regular?

My Understanding/Doubt
We can say that Language

$L_{2}=\left \{ wxw \,\,|\,\,w,x \,\in  \left ( a+b \right )^{*}\right \}$ is regular

By making  $x=(a+b)^{*}$ and $w=\epsilon$
Based on above Understanding , i can say that $L$ is regular?

My regular expression will be-:
$L=a(a+b)^{+}a+b(a+b)^{+}b+a(a+b)^{+}b+b(a+b)^{+}a$
Am i right?
When is a string of the form $a^*ba^*b$ in the language $L$, i.e., of the form $wxw$, for nonempty $w,x$?

Or, to be more explicit, what is the language $L \cap a^*ba^*b$?