Number theory - why does the dot product on the Minkowski embedding resemble the Frobenius inner product?

The trace form $(a,b)\mapsto \mathrm{tr}(ab)$ is easily motivated as a choice of bilinear form on a number field $K$ by noting that it agrees with the (standard real) dot product on $\mathbb{R}^{r_1}×\mathbb{C}^{r_2}$ as restricted to the Minkowski embedding of $K$ (the one that sends a number to the vector of its images under the various real/complex embeddings - no normalising factors, as I gather is common). This directly justifies it's use in calculating discriminants of integral bases, viewed as fundamental volumes of the lattices they span in Minkowski space.

I have also seen it observed that it can be thought of as a Frobenius inner product, since we view elements of $K$ as linear transformations $K\to K$ (although the typical Frobenius product $(A,B)\mapsto \mathrm{tr}(A^TB)$ has that transpose kicking around*). I find this product an eminently obvious choice for generalising the dot product to matrices.

I suppose it's not surprising that this natural generalisation of the dot product agrees with the regular dot product in this situation where it there is some sense in which it can do so. However I am wondering if there is a more substantive reason for this to be the case (for example, whether the space of linear transformations $K\to K$ can be meaningfully related to Minkowski space). Anything to this effect would interest me greatly - thanks in advance.

*I am also curious to know if including the transpose would make a difference in the number field case, as I find it quite plausible that it would not. If it did, though, I must imagine this as having implications for the main question, as we would then have to explain the (more nebulous) formal similarity between these bilinear forms rather than their coinciding values.