# Problems for a concrete example in representation theory

I am currently studying representation theory. In my courses notes, I face trouble understanding one step of a concrete example: three masses connected by springs. This is the same approach as in Georgi p27 http://mural.uv.es/rusanra/Lie%20Algebras%20in%20Particle%20Physics%202%C2%AA%20ed%20-%20From%20Isospin%20to%20Unified%20Theories%20(Georgi,%201999).pdf

My problem happen when we have to find the normal modes associated to $$D_2$$. This point is not fully developped in Georgi. However, in my courses notes, I find this explanation:

"Since $$D_6 = D^{def} \otimes D_2$$ and that $$D^{def} = D_1 \oplus D_2$$, we have $$D_6 = [D_1 \oplus D_2] \otimes D_2 = (D_2 \otimes D_2) \oplus (D_2 \otimes D_2) = D_2 \oplus ( D_2 \otimes D_2)$$.

We can then see that the direction in $$\mathbb{R}^3$$ invariant under $$D^{def}$$ tensorized with $$\mathbb{R}^2$$ transforms under $$D_2$$."

From this fact, he deduce that $$(1,0,1,0,1,0)$$ for instance must be a normal mode. I understand all of that except the last sentence: Why is this direction transforming like $$D_2$$ ? ( I guess transforms under ... mean is an invariant subspace under ... ).

Thanks.