  We describe the slice spectral sequence of a 32-periodic $C_{4}$-spectrum
$K_{[2]}$ related to the $C_{4}$ norm ${N_{C_{2}}^{C_{4}}MU_{\bf R}}$ of the
real cobordism spectrum $MU_{\bf R}$. We will give it as a spectral sequence of
Mackey functors converging to the graded Mackey functor $\underline{\pi
}_{*}K_{[2]}$, complete with differentials and exotic extensions in the Mackey
functor structure.
  The slice spectral sequence for the 8-periodic real $K$-theory spectrum
$K_{\bf R}$ was first analyzed by Dugger. The $C_{8}$ analog of $K_{[2]}$ is
256-periodic and detects the Kervaire invariant classes $\theta_{j}$ in the
stable homotopy groups of spheres. A partial analysis of its slice spectral
sequence led to the solution to the Kervaire invariant problem, namely the
theorem that $\theta_{j}$ does not exist for $j\geq 7$.
