Definition of Euler system of cyclotomic units

Question

I am not sure about my understanding of Euler system of cyclotomic unit. This is what I have learnt:

Let $F=\mathbb{Q}(\mu_m)$. Let $\mathcal{I}(m)$ = {positive square free integers divisible only by primes $\mathit{l}\equiv{1}$ (mod $m$)}.

Let $\mathit{r}\in\mathcal{I}(m)$ i.e $\mathit{r} = \mathit{l_1}\mathit{l_2}...\mathit{l_r}$ such that $\zeta_r = \prod\limits_{l\mid r} \zeta_l$, then

An Euler system over the field $\mathbb{Q}(\mu_m)$ is defined to be a map $\alpha\colon\mathcal{I}(m)\rightarrow\overline{\mathbb{Q}}^\times$ such that $\forall\mathit{r}\in\mathcal{I}(m)$ and each prime $\ell|\mathit{r}$ we have:

1. $\alpha(r)$ = $\prod\limits_{j}\left((1-\zeta_m^j\zeta_r)(1-\zeta_m^{-j}\zeta_r)\right)^{a_j}$ $\quad$ $j,a_j\geqslant 1$ $\quad$ $\longrightarrow$ $\in\mathbb{Q}(\mu_{mr})^\times$ = $F(\mu_r)^\times$

2. $\alpha(r)$ is a unit

3. if $\ell\nmid r$ then $\quad$ $N_{F(\mu_{rl})/F(\mu_r)}\alpha(rl)$ = $\alpha(r)^{\mathrm{Frob}_\ell - 1}$

4. $\alpha(r\ell)\equiv \alpha(r)$$\quad$ modulo all primes of $F(\mu_{r\ell})$ above $\ell$.

How could this be related to what Rubin explains: Rubin explains that if we fix a collection $\{\zeta_m\colon m\geqslant1\}$ such that $\zeta_m$ is a primitive $m$-th root of unity and $\zeta_{mn}^n$=$\zeta_m$$\quad$$\forall{m,n}$ then for any $m\geqslant1$ and prime $\ell$ we have the relation:

$N_{\mathbb{Q}(\mu_{ml})/\mathbb{Q}(\mu_m)}(1-\zeta_{ml})$ =

a) $1-\zeta_m$, if $l\mid m$

b) $(1-\zeta_m)^{\mathrm{Frob}_l-1}$, if $l\nmid m$

Now we define

$\tilde{C}_{m,p} = N_{\mathbb{Q}(\mu_{mp})/\mathbb{Q}(\mu_m)} (1-\zeta_{mp})\in\mathbb{Q}(\mu_m)^\times$

and

$\tilde{C}_m = N_{\mathbb{Q}(\mu_m)/\mathbb{Q}(\mu_m)^+}(\tilde{C}_{m,p})$

Then the distribution relation above makes the collection $\{\tilde{C}_{m,p},\tilde{C}_m\}$ into an Euler System.