26

GEORGE A. HAGEDORN

/

5

E bJxi

Ni(X)

=

0

E

hixi

c2X2 + ^3X3 e

4

X

4

+ 1/5X5

-e

4

X

4

+ i/

5

X

5

c

2

X

2

- id5X3

\

\

c

2

X

2

- Jd

3

X

3

- e

4

X

4

- 1/5X5

e

4

X

4

- 1/5X5 c

2

X

2

+ 1^3X3

-E6^'

- E bix

(2.7)

/

To prove these claims we first note that if we replace ipj(X) by ipj(X), where

^P0 = /C^i(X),

MX) = z3ij3(x) +

Z^MX),

MX) = /c,^3W,

with H 2 + |2:2|

with |^3|2 + k

4

| 2 = 1, and

then N\(X) is transformed into

/ b-X

Ni(X) =

0

0 b-X

c-X-id-X -l-X-i] -X

\e-X-i~f-X c-X + id-X

c-X + id-X e-X + i~f -X\

-e - X H- i / • X c-X-id-X

-b-X 0

0

-b-X J

We show below that by making an appropriate choice of the ZJ, we can force c, d, e, and

/ to be mutually orthogonal (and all non-zero in generic situations). Once this is done, we

rotate the X2, X3, X4, and X5 coordinate axes, so that c, 5, e, and / point along the X2,

X3, X4, and X5, respectively. This proves the claims.

Arbitrarily choosing the z^'s is equivalent to arbitrarily choosing two matrices U\ G

677(2) and U2 € 677(2), so that

(MX)\

=

(MX)\

\fc(x)J

l

W2P0;

and

(h{x)\

\MX))

u

(^x)\