Invariant Polynomials on $\mathfrak{gl} (r,\mathbb R)$

Question

My book is Connections, Curvature, and Characteristic Classes by Loring W. Tu (I'll call this Volume 3), a sequel to both Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott (Volume 2) and An Introduction to Manifolds by Loring W. Tu (Volume 1).

I refer to Section B.1 (part 1), Section B.1 (part 2), Section B.3 (part 1),Section B.3 (part 2), Section 23.1 (part 1) and Section 23.1 (part 2) and to Invariant Polynomials on $\mathfrak{gl} (r,F)$. These look like Tu's way of introducing invariance to an audience unfamiliar with group actions.

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Question: For the paragraph before Example 23.1, what exactly does it mean that to say that $(23.1)$ "holds when $X$ is a matrix of indeterminates"?

Context: I'm asking this in relation to

1. the "i.e." part after.

   • 1.1. I think the "i.e." part is already from the assumption "$(23.1)$ holds for all $r \times r$ matrices $X$ of real numbers."

2. the use of Proposition B.1

   • 2.1. Perhaps this is supposed to be Proposition B.5 instead.

   • 2.2. Proposition B.1 is used twice more in Section 23.1 (part 2). I'm actually wondering about the relevance of Proposition B.1 in all three uses, but I think I cannot ask about Proposition B.1 without first understanding "holds when $X$ is a matrix of indeterminates".