  The tensor product $K \mathop{\otimes_{\cal R}} C_2'$ of the
${}^*$-continuous Kleene algebra $K$ with the polycyclic ${}^*$-continuous
Kleene algebra $C_2'$ over two bracket pairs contains a copy of the fixed-point
closure of $K$: the centralizer of $C_2'$ in $K \mathop{\otimes_{\cal R}}
C_2'$. We prove a representation of elements of $K\mathop{\otimes_{\cal R}}
C_2'$ by automata \`a la Kleene and refine it by normal form theorems that
restrict the occurrences of brackets on paths through the automata. This is a
foundation for a calculus of context-free expressions. We also show that $C_2'$
validates a relativized form of the ``completeness property'' that
distinguishes the bra-ket ${}^*$-continuous Kleene algebra $C_2$ from the
polycyclic one.
