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What are examples of toral non-abelian Lie subalgebras? This is toral because the adjoints of all its elements are semisimple A way to see it is to note that each $\operatorname {ad} (x)$ acts as a 3x3 skew-symmetric matrix, and is thus semisimple At the same time, $\mathfrak {su} (2)$ is obviously not abelian, hence it fits the prescription
lie algebras - Maximal toral $\iff$ toral and maximal abelian . . . So [$\mathfrak h$ is maximal toral $\impliedby$ $\mathfrak h$ is toral and maximal abelian] is true But how can I prove [$\mathfrak h$ is maximal toral $\implies$ $\mathfrak h$ is toral and maximal abelian]?
Abelianity of Toral Lie subalgebras - Mathematics Stack Exchange I'm a bit confused about the abelianity of the Toral lie subalgebras and the related discussion reported here Preliminarily I set a notation and recall some results: Let $\\mathfrak{g}$ be Lie alg
representation theory - Cartan subalgebras of classical Lie algebras . . . As final words of warning, as noted in a comment, a) these algebras, even when written exactly as matrices as above, contain many other Cartan subalgebras which are not given by diagonal matrices; and b) note that at least over certain base fields, one can alternatively represent the Lie algebras by matrices which do not even contain non-trivial diagonal ones E g $\mathfrak {so}_3 (\mathbb C
Is a toral subalgebra of semisimple Lie algebra, which has dimension . . . Consider a real semisimple Lie algebra $\mathfrak {g}$ and a toral subalgebra $\mathfrak {h}$, which in this case is equivalent to being abelian and made of ad-semisimple elements (i e they are ad-diagonalizable over the complex field)
Any hyperbolic automorphism of $\mathbb {T}^n$ is mixing. We know a hyperbolic toral automorphism is defined to be an automorphism of $\mathbb {T}^n=\mathbb {R}^n \mathbb {Z}^n$ induced by integer matrix in $\text {GL} (n,\mathbb {Z})$ which has no eigenvalue of module $1$