h = {H (a, d) = diag (a, d, - a, - d, 0) : a, d \in C} .

Put

ε_{1} = L_{1}

and

ε_{2} = L_{2}

; then

L_{3} = - ε_{1}

L_{4} = - ε_{2}

and

L_{5} = 0

h

E_{ij}

[H, E_{ij}] = (L_{i} - L_{j}) (H) E_{ij}

. Applying this to the general matrix in part (i) gives the following one-dimensional root spaces and chosen root vectors:

α ε_{1} - ε_{2} - ε_{1} + ε_{2} ε_{1} + ε_{2} - ε_{1} - ε_{2} ε_{1} - ε_{1} ε_{2} - ε_{2} X_{α} E_{12} - E_{43} E_{21} - E_{34} E_{14} - E_{23} E_{32} - E_{41} E_{15} - E_{53} E_{35} - E_{51} E_{25} - E_{54} E_{45} - E_{52}

Each row satisfies

[H, X_{α}] = α (H) X_{α}

so_{5} = h \oplus α \in R ⨁ C X_{α}, R = {\pm ε_{1}, \pm ε_{2}, \pm ε_{1} \pm ε_{2}} .

This is the B2 root system, with short roots on the coordinate axes and long roots on the diagonals.

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