# Multiple imputation and normality assumption

I am reading the an online book by Stef Van Buuren (link at bottom) regarding multiple imputation. In Section 3.2.1 he lists 4 different approaches to multiple imputation:

Later on in Section 3.3 he states: "The imputation methods discussed in Section 3.2 produce imputations drawn from a normal distribution."

I'm struggling to see how this is the case. Obviously I can see the error term added is drawn from a normal distribution. But surely the distribution of the imputations depend on the distribution of each of the predictor variables in X? Seeing as it is a linear regression, I would have thought the distribution of Y would be a combination of all of the distributions of the X's, as well as the error term.

SciKit learn's IterativeImptuter documentation also states: "The version implemented assumes Gaussian (output) variables. If your features are obviously non-normal, consider transforming them to look more normal to potentially improve performance"

I feel like this is because the default option for IterativeImputer draws the regression coefficients from a gaussian posterior distribution.

Based on all of the above I have the following questions:

1. Where in the above equations is normality assumed, and why do the equations "produce imputations drawn from a normal distribution".
2. Is the distribution of any imputed value determined by the model/equations used, which surely themselves depend on the distribution of X's, which are not necessarily normal?

Stef Van Buuren book: https://stefvanbuuren.name/fimd/sec-linearnormal.html