 Let us describe the transformations of $\R^4$ given by each generator. Consider a new base $(e_1, e_2,
e_3, e_4)$, given by $e_1=f_1+f_2$, $e_2=f_1-f_2$, $e_3=f_3+f_4$, $e_4=f_3-f_4$. The generator $r$ of order 4 is represented by the rotation in the plane $(e_2,e_4)$ through the angle $\frac{\pi}{2}$ and the reflection in the plane $(e_1,e_3)$ with respect to the line $e_1+e_3$. The generator $l$ of order 2 is represented by the central symmetry in the plane $(e_1, e_3)$.

Obviously, the described representation of $\I_4$ admits invariant (1,1,2)-dimensional subspaces. We will denote subspaces by $\lambda_1, \lambda_2, \tau$.

The lines $\lambda_1, \lambda_2$ are generated by the vectors $e_1+e_3$, $e_1-e_3$ correspondingly. The subspace $\tau$ is generated by the vectors $e_2, e_4$. The generator $r$ acts by the reflection in $\lambda_2$ and by the rotation in $\tau$ throught the angle $\frac{\pi}{2}$. The generator $l$ acts by reflections in the subspaces $\lambda_1$, $\lambda_2$.

In particular, if the structure group $\Z/2 \int \D_4$ of a 4-dimensional bundle $\zeta: E(\zeta) \to L$ admits a reduction to the subgroup $\I_4$, then the bundle is decomposed into the direct sum $\zeta = \lambda_1 \oplus \lambda_2 \oplus \tau$ of $1,1,2$–dimensional subbundles.

### Definition 6 {#definition-6 .unnumbered}

Let $(g: N^{n-2k} \looparrowright \R^n, \Xi_N, \eta)$ be an arbitrary $\D_4$-framed immersion. We shall say that this immersion is an $\I_b$–immersion (or a cyclic immersion), if the structure group $\Z/2 \int \D_4$ of the normal bundle over the double points manifold $L^{n-4k}$ of this immersion admits a reduction to the subgroup $\I_4 \subset \Z/2 \int \D_4$. In this definition we assume that the pairs $(f_1, f_2)$, $(f_3,f_4)$ are the vectors of the framing for the two sheets of the self-intersection manifold at a point in the double point manifold $L^{n-4k}$. $$$$

In particular, for a cyclic $\Z/2 \int \D_4$-framed immersion there exists the mappings $\kappa_a: L^{n-4k} \to K(\Z/2,1)$, $\mu_a: L^{n-4