If two Lie groups $G_1 \cong G_2$, how can their minimal representations be complex and pseudoreal, differently?

Question

Here are some facts:

• $SU(4) = Spin(6)$

• The fundamental representation of $SU(4)$ is 4-dimensional complex, because it is not equivalent to its complex conjugate representation.

• The minimal representation of $Spin(6)$ is 4-dimensional pseudoreal, because it is equivalent to its complex conjugate representation, but there is no change of basis that makes all the matrix entries real.

How can the 4-dimensional $SU(4)$ representation be complex but the 4-dimensional $Spin(6)$ be pseudoreal, even if $SU(4) = Spin(6)$?

Namely if two Lie groups $G_1 \cong G_2$, how can their minimal representations be complex and pseudoreal, differently?

Answer 1

Simple answer: they cannot. The groups are identical so their representations are identical.

Note there isn't a minimal representation, there are two: the two half spin representations of $\operatorname{Spin}(6)$ which are the same thing as the defining representation of $SU(4)$ and its dual (physics calls these the fundamental and anti-fundamental reps). This pair of reps are both complex and are dual to each other and conjugate to each other, so they are not real or pseudoreal.

Edit: I should probably also do the standard PSA that a lot of these terms are used differently in maths versus physics literature. I am assuming that you are coming from physics so that "real" means the complex representation descends to a real one, "pseudoreal" means there is a compatible quaternionic structure on the representation and "complex" means that neither of these hold.