John Stillwell - Naive Lie Theory.pdf - Index of

130 6 Structure of Lie algebras
[X,E ij ]=
⎛
i j
⎞
−x 1 j x 1i
−x 2 j x 2i
. .
i
−x j1 −x j2 ··· 0 ··· 0 ··· −x jn
.
. .
j
x i1 x i2 ··· 0 ··· 0 ··· x in
⎜
⎟
⎝
. .
⎠
−x nj x ni
(*)
We now make a series of applications of formula (*) for [X,E ij ] to
reduce a given nonzero X ∈ so(n) to a nonzero multiple of a basis vector.
The result is the following theorem.
Simplicity of so(n). For each n > 4, so(n) is a simple Lie algebra.
Proof. Suppose that I is a nonzero ideal of so(n), andthatX is a nonzero
n × n matrix in I. We will show that I contains all the basis vectors E ij ,so
I = so(n).
In the first stage of the proof, we Lie bracket X with a series of four
basis elements to produce a matrix (necessarily skew-symmetric) with just
two nonzero entries. The first bracketing produces the matrix X 1 =[X,E ij ]
shown in (*) above, which has zeros everywhere except in columns i and j
and rows i and j.
For the second bracketing we choose a k ≠ i, j and form X 2 =[X 1 ,E jk ],
which has row and column j of X 1 moved to the k position, row and column
k of X 1 moved to the j position with their signs changed, and zeros where
these rows and columns meet. Row and column k in X 1 =[X,E ij ] have
at most two nonzero entries (where they meet row and column i and j),
so row and column j in X 2 =[X 1 ,E jk ] each have at most one, since the