I am studying the transition rule of electric dipole transition. I have some question about some rules in the Wikipedia: https://en.wikipedia.org/wiki/Selection_rule#Summary_table
In that table, they claim that in the LS coupling when $\Delta s=0$, $\Delta l=0$ is one of allowed transitions for hydorgen-like atoms, and even in the intermediate coupling $\Delta l$ can also be 0 or even $\pm2.$
However, how can I get this conclusion?
I tried to calculate this:
For the LS coupling cases, if I express the states in terms of $|lm_l\rangle|s,m_s\rangle$ and express $\langle j',m'_j|er_q|j,m_j\rangle$ :
$$ |j,m_j\rangle=\sum_{m_s,m_l}\langle l,m_l;s,m_s|j,m_j\rangle|l,m_l;s,m_s\rangle $$
Then: ($q=0,\pm1$ $\pm1$ is for left/right circular polarization and 0 is for linear polarization.)
$$ \begin{split} \langle j',m'_j|er_q|j,m_j\rangle&=\sum_{m_l,m_s,m_l',m_s'}\langle l',m'_l;s',m'_s|er_q|l,m_l;s,m_s\rangle \langle j',m'_j|l',m'_l;s',m'_s\rangle \langle l,m_l;s,m_s|j,m_j\rangle\\ &= \sum_{m_l,m_s,m_l',m_s'} \langle l',m'_l|er_q|l,m_l\rangle \langle s',m_s'|s,m_s\rangle \langle j',m'_j|l',m'_l;s',m'_s\rangle \langle l,m_l;s,m_s|j,m_j\rangle\\ \xrightarrow{\langle s',m_s'|s,m_s\rangle=\delta_{ss'}\delta_{m_s,m_s'}}&=\sum_{m_l,m_l',m_s} \langle l',m'_l|er_q|l,m_l\rangle \langle j',m'_j|l',m'_l;s,m_s\rangle \langle l,m_l;s,m_s|j,m_j\rangle\\ \end{split} $$
If the electric dipole moment does not vanish then $\langle l',m'_l|er_q|l,m_l\rangle$ can not vanish so $\Delta l\neq0.$
According to the Wigner-Eckart theorem,
$$ \langle j',m'_j|er_q|j,m_j\rangle=\langle j',m'_j||er_q||j,m'_j\rangle \langle j,m_j;1,q|j',m'_j\rangle $$
The selection rule is:
$$ m'=m+q,\\ j-1\leq j'\leq j+1 $$
So according to this theorem the selection rule is: $\Delta j=0,\pm1,\ \Delta m=q.$ Even I pick $\Delta j=0$, I must still need to pick up two states whose $\Delta l\neq0.$ Also, how can I have two states whose $\Delta s=1$ in the intermediate coupling? Because as I know for hydrogen-like atoms there is only one valence electron so the spin angular momentum should always be $1/2$.
And how can two states with $\Delta l=\pm2$ be coupled in the intermediate coupling? And what does intermediate coupling means?