A paradoxical decomposition of a free group of rank two

For $x\in \{a,a^{-1},b,b^{-1}\}$ let $W(x)$ be the set of nonempty reduced words beginning with $x$, and put $P=\{a^n:n\ge 0\}$ and $P_{+}=\{a^n:n\ge 1\}$. Define $A_1=W(a)\setminus P_{+}$, $A_2=W(a^{-1})\cup P$, $B_1=W(b)$, and $B_2=W(b^{-1})$.

The four sets in step 1.1 are pairwise disjoint and partition $F(a,b)$: the nonidentity reduced words have one of the four possible first letters, and $P_{+}$ has been moved from $W(a)$ into the piece containing the identity.

Reduction of the first letter gives $aW(a^{-1})=F(a,b)\setminus W(a)$ and $aP=P_{+}$, hence $F(a,b)=A_1\bigsqcup a{A}_2$. Similarly $bW(b^{-1})=F(a,b)\setminus W(b)$, hence $F(a,b)=B_1\bigsqcup b{B}_2$. Therefore the pieces in step 1.1 with translators $e,a,e,b$ satisfy [L1] and form a paradoxical decomposition.