Thank you! I will take a look at the reference. Yes, the smallest reflex field is for the adjoint Shimura datum. The torus part is always mysterious to make a moduli description in Hodge-type cases.

Such a representation is often called visible (see Dadok–Kac and also arxiv.org/pdf/1604.04121 Lemma 4.1). Irreducibly faithful visible reps have been classified by Kac and Dadok–Kac (over complex numbers). In the irreducible case, many notions agree (see also arxiv.org/abs/1905.04845v4). You may be also interested in a similar question mathoverflow.net/questions/473836/….