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An invertible linear map $\varphi$ on a Lie algebra $L$ is called a triple automorphism of it if $\varphi \left(\right[x,[y,z]\left]\right)=\left[\varphi \right(x),[\varphi \left(y\right),\varphi \left(z\right)\left]\right]$ for $\forall x,y,z\in L$. Let $𝔤$ be a finite-dimensional simple Lie algebra of rank $l$ defined over an algebraically closed field $F$ of characteristic zero, $𝔭$ an arbitrary parabolic subalgebra of $𝔤$. It is shown in this paper that an invertible linear map $\varphi$ on $𝔭$ is a triple automorphism if and only if either $\varphi$ itself is an automorphism of $𝔭$ or it is the composition of an automorphism of $𝔭$ and an extremal map of order $2$.

Let $R$ be an arbitrary commutative ring with identity, $gl(n,R)$ the general linear Lie algebra over $R$, $d(n,R)$ the diagonal subalgebra of $gl(n,R)$. In case 2 is a unit of $R$, all subalgebras of $gl(n,R)$ containing $d(n,R)$ are determined and their derivations are given. In case 2 is not a unit partial results are given.

Let $𝒫$ be an arbitrary parabolic subalgebra of a simple associative $F$-algebra. The ideals of $𝒫$ are determined completely; Each ideal of $𝒫$ is shown to be generated by one element; Every non-linear invertible map on $𝒫$ that preserves ideals is described in an explicit formula.

Let $𝒩=N_n\left(R\right)$ be the algebra of all $n\times n$ strictly upper triangular matrices over a unital commutative ring $R$. A map $\varphi$ on $𝒩$ is called preserving commutativity in both directions if $xy=yx\iff \varphi \left(x\right)\varphi \left(y\right)=\varphi \left(y\right)\varphi \left(x\right)$. In this paper, we prove that each invertible linear map on $𝒩$ preserving commutativity in both directions is exactly a quasi-automorphism of $𝒩$, and a quasi-automorphism of $𝒩$ can be decomposed into the product of several standard maps, which extains the main result of Y. Cao, Z. Chen and C. Huang (2002) from fields to rings.