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I am not getting a step in simple inductive proof in Lie Algebra: it is below. Theorem 2.6 I have copies just for knowing hypothesis; for this, a proposition is used, and my question is about understanding a step in it.

What is not getting: It is defined that $$x_i=(ad y-\beta1)^ix\in L.$$ Then in the expansion of $r+1$th power, see the term after binomial coefficient: it is written $\rho(x_i)$. I am not understanding this; whether this should be $x_i$ or $\rho(x_i)$? This could be very easy, but I am new in Lie algebra. Please help me.

Thanks for the patience and in advance for help!

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    This is in Lie Algebras of Finite and Affine Type - Carter, page 17. – 2017-01-02
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    So is this correct: in the equation before proof of proposition, in LHS, we replace $x$ by $\rho(x)$ and also in RHS, we replace $x$ by $\rho(x)$; is this OK? – 2017-01-02
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    Note that $x_i$ acts by multiplication so really it is the same as $\rho $. This is a bit of a screwy thing that people in representation theory do but it's just repackaging notation, that's all. – 2017-01-02
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    I am partially getting your point; but is it ok what I said in my earlier comment? I mean, doing this correction, will it be ok? – 2017-01-02
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    Yes on the left hand side, right hand side is a bit less clear to me because of the $\operatorname{ad} $ mucking things up. You probably could but you'd have to lift the $\operatorname{ad} $ to $\rho$ in the obvious way. – 2017-01-02

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