Rank Two Approximations of $2 \times 2 \times 2$ Tensors over $\mathbb{R}$

We provide a coordinate-free proof that real $2 \times 2 \times 2$ rank three tensors do not have optimal rank two approximations with respect to the Frobenius norm. This result was first proved in by considering the ${\text{GL}(V^1) \times \text{GL}(V^2) \times \text{GL}(V^3)}$ orbit classes of ${V^1 \otimes V^2 \otimes V^3}$ and the $2 \times 2 \times 2$ hyperdeterminant. Our coordinate-free proof expands on this known result by developing a proof method that can be generalized more readily to higher dimensional $n_1 \times n_2 \times n_3$ tensor spaces.