Consider a simple Lie group $G$ and its mod $p$ cohomology $H^*(G, \mathbb{Z}_p)$.

A good reference is the book

Mimura, Mamoru; Toda, Hirosi, Topology of Lie groups, I and II. Transl. from the Jap. by Mamoru Mimura and Hirosi Toda, Translations of Mathematical Monographs. 91. Providence, RI: American Mathematical Society (AMS). iv, 451 p. (1991). ZBL0757.57001.

Consider the diagonal map $\Delta: H^*(G, \mathbb{Z}_p) \rightarrow H^*(G, \mathbb{Z}_p) \otimes H^*(G, \mathbb{Z}_p)$. Let $x_i$ for $i=1,...,n$ denote the generators of $H^*(G, \mathbb{Z}_p)$. Assume there is an element $x^a_i\otimes x^b_j$ contained in $\Delta(x_m)$, with $a,b \neq 0$.

Question 1: Is it true that at least one of the numbers $a, b$ is equal to $1$?

Question 2: Is it true that an element $x_ix_j \otimes x_l$ or $x_i \otimes x_jx_l$ can never be contained in $\Delta(x_m)$?

I assume that both answers are "yes", after checking many examples calculated in the mentioned book and also a paper on this topic released by Mimura and Kono.

The second question basically asks, wether the diagonal map can "mix" generators.

In the book the diagonal map is named $\phi$ and $\phi'(x):= \phi(x) - 1\otimes x - x\otimes 1$. The results are presented in terms of $\phi'$, which shouldnt change anything.