vzp={1,0,0};vz0={0,1,0};vzm={0,0,1};

Lz=h{{1,0,0},{0,0,0},{0,0,-1}};Lz//MatrixForm

Lx=h/Sqrt[2]{{0,1,0},{1,0,1},{0,1,0}};Lx//MatrixForm

Ly=h/Sqrt[2]/I{{0,1,0},{-1,0,1},{0,-1,0}};Ly//MatrixForm

L2=Lx.Lx+Ly.Ly+Lz.Lz; L2//MatrixForm

Lz.vzp

Lz.vz0

Lz.vzm

Eigensystem[Lx]

vxp={1,(Sqrt[2]),1}/Sqrt[{1,(Sqrt[2]),1}.{1,(Sqrt[2]),1}]

vxm={1,-(Sqrt[2]),1}/Sqrt[{1,-(Sqrt[2]),1}.{1,-(Sqrt[2]),1}]

vx0={-1,0,1}/Sqrt[{-1,0,1}.{-1,0,1}]

vxp.vz0

vx0.vz0

vxm.vz0

vxp.vzp

vx0.vzp

vxm.vzp

vxp.vzm

vx0.vzm

vxm.vzm

Lx.vxp == h vxp

Lx.vxm == -h vxm

Lx.vx0 == 0 vx0

Lx.Ly-Ly.Lx == I h Lz