Not sure if this asnwers your question: regard the factors as vectors (in an n-dimensional space). Two vectors are orthogonal ("normal") when their scalar product is zero. Also, any factor must be linearily independent of all other vectors.

Factor analysis is a multivariate method therefore assumes that the variables are continuous and normal distribution assumptions, there really are some more, then ensures that the factors have normal distribution. This is seen as a mathematical statement. But, in practice, through the accumulation of much empirical evidence has established its relevance to establish the construct validity of a set of items with a Likert scale, for example. But in this case, there is no guarantee that the obtained factors present a normal distribution. However, there are nonparametric statistical methods that do not require a normal distribution. Furthermore, verify the normality of the factors extracted before applying parametric statistical method, it is a practice that distinguishes the most rigorous and credible research.

In factor analysis, we need to check the normality of data because normality is one of the pre-requisites of the analysis.

Thank you all for sharing your knowledge .