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