Is including the key as AAD actually dangerous? In a recent answer, Maarten mentioned 


  including the key is a bad idea as well.


This got my curious. Is there any scientific / cryptographic analysis whether including the key in the authentication actually allows for (easier) attacks in the standard model(s) for authenticated encryption?

Obviously this means that both sides include the key in the AAD but don't send it in clear.

I also see that it's kinda pointless to include it, but is it actually dangerous? (less formal than above)

If no generic answer is possible, please assume AES-GCM as the scheme in question
One potential issue with GCM is that it can potentially make the problems you get from repeating nonces worse; instead of allowing you to forge, and revealing the plaintext for the packets with the repeated nonces, it can become a key recovery issue.

Here's one way it can happen; suppose the AAD is the 128 bit key, and you repeat the nonce with three different messages of different lengths (a 1 block, a 2 block and a 3 block message).  Then, the three GCM tags are computed as:

$$KH^3 + C_1H^2 + C_2H + Y = T_1$$
$$KH^4 + C_3H^3 + C_4H^2 + C_5H + Y = T_2$$
$$KH^5 + C_6H^4 + C_7H^3 + C_8H^2 + C_9H + Y = T_3$$

where $K$ is the unknown key, $H$ is the unknown $H = E_k(0)$ value, $Y$ is the unknown value $Y = E_k(nonce)$ used to protect the tag (and because all three messages use the same nonce, they have that value in common), and $C_1, ..., C_9$ and the tag values $T_1, T_2, T_3$ are values known to the attacker.

These are three polynomial equations in three unknowns; they can be combined into a single fourth degree polynomial in $H$ (and not $K$ or $Y$); it can be solved for $H$ (giving up to 4 possible solutions), from that, you can recover the secret key $K$ value as one of 4 possibilities.