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

A good reference is the book

"Topology of Lie groups", by Mimura and Toda.

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 is $a, b$ 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.