For a finite group it's nothing special if two onedimensional irreps pop up in a product, e.g. for $C_{3v}$ symmetry, $E\bigotimes{E}=A_1\bigoplus{A_2}\bigoplus{E}$ or in dimensions, $2*2=1+1+2$. Schur's Lemma merely forces that only one $A_1$ can be in the product.
Now I never saw a Lie group irrep with dimension $1$ which is not the trivial irrep. Have I just not looked around far enough? :) (Maybe e.g. beyond semisimple ones?)
(Background: Playing around with tensors for 3nodes and crossings, I found a sixdimensional irrep which gives an invariant for tangled graphs, the ClebschGordan series being $6*6=1+1+6+8+8+12$. The second $1$ is kind of an "antisymmetric one", like $A_2$. In every respect (I could check), this invariant behaves like the invariants coming from Lie groups.)
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