![Given that CAB, CJ = ACB, CJ + [AC]B we have to show that [ 1², L₂J :o (L2, L₂] = [lukt lý + 22², L2] & Chh, L2] + [lŷ , [2]](//img.homeworklib.com/questions/27fcd450-81ca-11eb-939b-538e2e978194.png?x-oss-process=image/resize,w_560)
![Page No. Le Cly, 2nd + ln [ln, Lallyt de [ n, l. n lnly Since [ly, Lu] = alth Lz (2n, ru ] zo [ & Ly, Lx] = 2 m (- 8h22) + ln](//img.homeworklib.com/questions/28d81410-81ca-11eb-8990-ffeb07a786d8.png?x-oss-process=image/resize,w_560)
![[ly, un] = - ith Ly L2 - it Lylzly - 10th 22 lg We have to compute the Commututor (hely, Lu] Ly, Ln] = 12 [Ly , Ln] + [ 1² Le](//img.homeworklib.com/questions/2ac061a0-81ca-11eb-874e-4d9283385a96.png?x-oss-process=image/resize,w_560)
![= [ln Ly, Lx] + (LJ, Ln] + [h Ly, Lu] putting value of Clin Ly, Lu] from 12 FCW Lu] from (21 and [ 12 Ly , and from (4) [ 2²](//img.homeworklib.com/questions/2b898150-81ca-11eb-9080-0196a91742cc.png?x-oss-process=image/resize,w_560)
Given that CAB, CJ = ACB, CJ + [AC]B we have to show that [ 1², L₂J :o (L2, L₂] = [lukt lý + 22², L2] & Chh, L2] + [lŷ , [2] + [ LZ , L2] ده لعابداع + [علیجاگیا۔ العليوا خوا [Ly, L2] Ly & L2 [ 12 L 2] + CL 2 , L2] 22 As Ln, LZ) = -ctly [Ly, L2J ihla CL2,12) =0 [2², L₂] = La citly) + (1²hLy] La it Ly Lietln) + Litlu) Ly + L2 (0) + (0) mit Ly Ly - it Lyle tih Lyle tehny So [ L², L₂J 20 we have to compute the commutator [urly, Ln] [LuLy , Lz] - [h[Ly, Lu] & [h, Lally- we have med CAB, C = A[B, CJ +(A, 26
Page No. Le Cly, 2nd + ln [ln, Lallyt de [ n, l. n lnly Since [ly, Lu] = alth Lz (2n, ru ] zo [ & Ly, Lx] = 2 m (- 8h22) + ln (0) Ly + = - it in iz Ir taily, bz) : -et Lube to) Zri Ly 3. We have to - Commutator compute the [ 13, 2x] [L}, Ln] = [ Ly Ly, and - lý (Ly, Lm] + ll}, La Ily. = Lylly, lntlly Ly, Ln] Ly Ly [Ly, Lon] + Ly [Ly, In] Ly t - [ly, hn] Ly Ly Sincez tý Since [ly , Lon] - it lz [Ly, Lx] = Lý (-11 h L2) + Ly ( 19 h Lz) Ly + (rit Lz) L - it by L2 + - it Ly Lzly - it L z LZ
[ly, un] = - ith Ly L2 - it Lylzly - 10th 22 lg We have to compute the Commututor (hely, Lu] Ly, Ln] = 12 [Ly , Ln] + [ 1² LeJ Ly - = LZ [Ly, Ly] + L2 [ L2, Ln] Ly + [ L2, Luis I AS (Ly ; Ln] = -8 h Lz L [22, LM] it Ly [ 12 Ly , Lond: L2 (-it Lz) + L2 Lit Ly Ly tih Ly 22 Ly - eh 17 Lz tit L2 Ly + 1h Ly La Ly 11 22 Ly, Lx] = -lit L² + 1h Lalytit Ly laze 5. we have to compute the commutator [2 Ly , La] - [han thy the Ly, in [hn Ly, Ln] + [Lý Ly Ln] + [L² ly, in)
= [ln Ly, Lx] + (LJ, Ln] + [h Ly, Lu] putting value of Clin Ly, Lu] from 12 FCW Lu] from (21 and [ 12 Ly , and from (4) [ 2² Ly, Load = -1h Ln Lz - ch LÝ L2 - et Ly Loty - vħ Laly - ħ L² t con Laky + 10h Lyraly = -1°h Ln L2 - let ý lz - ceh L2 = -18h (Lu Lz thy Lz + 22 22) = -10th (in th} +22) Lz - - lt L² Lz { L² Ly, Ln] = -1°t