Closed form posterior for product of inverse Gamma and Normal distribution

I am currently reading a book about mixture analysis, and in the textbook a posterior for the parameters $$\mu1,\mu2,\sigma_1^2,\sigma_2^2$$ of a two-component gaussian mixture is derived as follows (S and y are the classifications and data and are given):

$$p(\mu1,\mu2,\sigma_1^2,\sigma_2^2|S,y)=p(\mu_1|\sigma_1^2,y,S)p(\sigma_1^2|y,S)\cdot p(\mu_2|\sigma_2^2,y,S)p(\sigma_2^2|y,S)$$

where $$p(\mu_1|\sigma_1^2,y,S)$$ and $$p(\mu_2|\sigma_2^2,y,S)$$ are normally distributed with some parameters, which I omit for readability here, and $$p(\sigma_1^2|y,S)$$ and $$p(\sigma_2^2|y,S)$$ are inverse gamma distributed with some parameters, which I also omit for readability here.

Now my question is: The author states that this is a closed-form solution for the joint posterior $$p(\mu1,\mu2,\sigma_1^2,\sigma_2^2|S,y)$$. I cannot recognize this product of two subproducts as a known distribution. I also checked the Wikipedia page for conjugate priors, but I could not find anything.

Can someone tell me if I'm overlooking something here?

I already derived the full conditional distributions, so Gibbs sampling would be possible, but this is not a closed-form solution then anymore.