# Lie algebras assignments

I need to solve two assignments in Lie algebras. These assignments are not very difficult, but my knowledges in Lie algebras aren't good.
1. Let $$\delta$$ be a derivation of the Lie algebra $$\Im$$. Show that if $$\delta$$ commutes with every inner derivation, then $$\delta$$($$\Im$$)$$\subseteq$$C($$\Im$$), where C($$\Im$$) denotes the centre of $$\Im$$ .

2. Let x $$\in$$ gl(n,F) have n distinct eigenvalues $$\lambda$$1..$$\lambda$$n in F. Prove that eigenvalues of ad$$_{}x$$ are the n$$^{}2$$ scalars $$\lambda$$$$_{}i$$-$$\lambda$$$$_{}j$$ (1$$\leq$$i,j$$\leq$$n)

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1. Let $$\delta$$ be a derivation of the Lie algebra $$\Im$$. Show that if $$\delta$$ commutes with every inner derivation, then $$\delta$$($$\Im$$)$$\subseteq$$C($$\Im$$), where C($$\Im$$) denotes the centre of $$\Im$$ .
You need to show that for all x and y, $$\delta(x)$$ commutes with y, i.e. $$[y,\delta(x)]=0$$. What's a central derivation? Can you rewite the condition using one?
2. Let x $$\in$$ gl(n,F) have n distinct eigenvalues $$\lambda$$1..$$\lambda$$n in F. Prove that eigenvalues of ad$$_{}x$$ are the n$$^{}2$$ scalars $$\lambda$$$$_{}i$$-$$\lambda$$$$_{}j$$ (1$$\leq$$i,j$$\leq$$n)