Difference between the pairings $\text{Tr}(xy)$ and $\text{Tr}(x^t y)$

Question

Let $\mathfrak{g}$ be the tangent space to $GL_n(\mathbb{C})$ at the identity.

What is the difference between the two maps? Any subtle geometric or algebraic difference between the two pairings

$$ \mathfrak{g}\times \mathfrak{g}\rightarrow \mathbb{C}, \hspace{4mm} (x,y)\mapsto \text{Tr}(xy) $$ versus

$$ \mathfrak{g}\times \mathfrak{g}\rightarrow \mathbb{C}, \hspace{4mm} (x,y)\mapsto \text{Tr}(x^t y), $$

where $x^t$ denotes the transpose of $x$ and $\text{Tr}(x)$ denotes the trace of the matrix $x$?

Answer 1

Probably the most important difference is that the first one is invariant under conjugation by an arbitrary matrix $P$, while the second is only invariant by conjugation under orthogonal maps $P^t P = I$.