Problem $1$ Algebra of angular moment operators

Please show the commutation relation between different angular moment operators.

$(20$ pts$)[widehat(L{)}_i,$widehat$(L{)}_j]=i^{}lo{n}_{\text{i}\text{j}\text{k}}$widehat$(L{)}_k$ with $i,j,$kin$\{x,y,z\}.$

$(20pts)[widehat(L{)}_z,$widehat$(L{)}_{+-}]=+-widehat(L{)}_{+-}$

$(20pts)[widehat(L{)}^2,$widehat$(L{)}_i]=0$ for all iin$\{x,y,z\}$

$(20pts)[widehat(L{)}^2,$widehat$(L{)}_{+-}]=0.$

The raising and lowering operators fulfill

widehat$(L{)}_{+}|{}_m$:$||$

$(20pts)$ Please show that $A_m=B_m^{**}$