Define commutator of operators, to do computations

As already discussed in the comments under OPs question: computations of this kind can be facilitated by some packages. Regarding the specific problem posed by OP a minimal semi-automatic solution would be

(* Check if a symbol is non-commutative under NCM: defaults False *)
ClearAll[NCQ]
NCQ[a__]:=NCQ@List[a];
NCQ[a_List]:=NCQ[#]&/@a
NCQ[Subscript[a_,i_]]:=NCQ[a]
NCQ[a_]:=False;

(* Non-commutative multiply *)
ClearAll[NCM];
NCM[]:=1;
NCM[a_]:=a
NCM[x___,NCM[y1_,y2__],z___]:=NCM[x,y1,y2,z]
NCM[x___,f_[y1_,y2___],z___]/;MemberQ[{Plus,List,Equal,Rule,RuleDelayed},f]:=f@@Map[NCM[x,#,z]&,{y1,y2}]
NCM[x___,Times[y1_,y2__],z___]:=NCM[x,y1,y2,z]
NCM[x__/;MemberQ[NCQ[x],False](*contains commuting factors*)]:=Module[{s,nc},
    s=1;
    nc=If[NCQ[#],#,s=Times[s,#];Nothing]&/@{x};
    Return[Times[s,NCM@@nc]]
]
NCM/:MakeBoxes[NCM[a_,b__],StandardForm]/;BoxForm`UseIcons:=ToBoxes@Row[{a,b},"**"]

(* Dagger symbol *)
ClearAll[dagger];
dagger/:MakeBoxes[dagger[a_],StandardForm]/;BoxForm`UseIcons:=ToBoxes@Superscript[a,"[Dagger]"]
dagger/:NCQ[dagger[a_]]:=NCQ[a]

(* Commutator symbol *)
ClearAll[NCC];
NCQ[NCC[a_,b_]]:=True;
NCC[a_,a_]:=0
NCC/:MakeBoxes[NCC[a_,b_],StandardForm]/;BoxForm`UseIcons:=ToBoxes@Row[{"[",a,",",b,"]"}]

followed by the explicit computation based by "normal" ordering non-commutative products (the method order) of as assuming $omega_1$, $omega_2$, $g$ and $bar{g}$ are commutative constants:

(* Establish a as a non-commuting symbol in NCM *)
NCQ[a]=True;

(* Setup normal-ordered commutators *)
NCC[Subscript[a, k_],dagger[Subscript[a, l_]]]:=KroneckerDelta[k,l]
NCC[dagger[Subscript[a, l_]],Subscript[a, k_]]:=-NCC[Subscript[a, k],dagger[Subscript[a, l]]]
NCC[Subscript[a, k_],Subscript[a, l_]]:=-NCC[Subscript[a, l],Subscript[a, k]]/;(!OrderedQ[{k,l}])
NCC[dagger[Subscript[a, k_]],dagger[Subscript[a, l_]]]:=-NCC[dagger[Subscript[a, l]],dagger[Subscript[a, k]]]/;(!OrderedQ[{k,l}])

(* Implement a "normal order" for products of a_i and dagger[a_i]*)
ClearAll[order]; 
order[exp_]:=Fold[ReplaceRepeated,exp,{
HoldPattern[NCM[x___,dagger[Subscript[a, k_]],Subscript[a, l_],y___]]:>NCM[x,Subscript[a, l],dagger[Subscript[a, k]],y]+NCM[x,NCC[dagger[Subscript[a, k]],Subscript[a, l]],y],
HoldPattern[NCM[x___,Subscript[a, k_],Subscript[a, l_],y___]]:>NCM[x,Subscript[a, l],Subscript[a, k],y]+NCM[x,NCC[Subscript[a, k],Subscript[a, l]],y]/;(!OrderedQ[{k,l}]),
HoldPattern[NCM[x___,dagger[Subscript[a, k_]],dagger[Subscript[a, l_]],y___]]:>NCM[x,dagger[Subscript[a, l]],dagger[Subscript[a, k]],y]+NCM[x,NCC[dagger[Subscript[a, k]],dagger[Subscript[a, l]]],y]/;(!OrderedQ[{k,l}])
}];

(** Computation **)
Subscript[[Omega],1]* NCM[Subscript[a,1],dagger[Subscript[a,1]]]+Subscript[[Omega],2]*NCM[Subscript[a,2],dagger[Subscript[a,2]]]+g*NCM[dagger[Subscript[a,1]],Subscript[a,2]]+gb*NCM[dagger[Subscript[a,1]],Subscript[a,2]]
NCM[%,Subscript[a, 1]]-NCM[Subscript[a, 1],%]
%//order//Expand//Simplify
%/.HoldPattern[NCC[Subscript[a, k_],Subscript[a, l_]]]:>0

which I simplified further in the last step assuming $[a_i,a_j]=0$, which if I recall Second quantization correctly should be the case. I have neither checked the code thoroughly nor did I do the computation by hand carefully, so I can not really guarantee it to be correct but it seems plausible to me apart from the strange definition of the last two terms in $H$ (I would have expected by looking at for something like $a_2^dagger a_1$ in the last term instead of being identical to the second to last apart from the prefactor).

This code works by first implementing the rudimentary method NCM for non-commutative products with the auxiliary function NCQ checking symbols for potentially non-commutative character. dagger and NCC are symbols for Hermitian conjugate (just a wrapper here) and symbolic commutators respectively. To perform the actual computation $[H,a_1]$ I implemented the explicit commutators for $a_i$ and $a_i^dagger$ and established a "normal ordering" with the rules in order. order commutes all daggers to the right and then orders all daggered and non-daggered appearances of $a$ according to Mathematicas canonical ordering.

This code is meant to given an impression on how one could implement such computations in an in principle straight-forward manner: which means that the code is probably terribly inefficient BUT most likely faster than a human. Implementing $[a_i,a_j]=0$ together with the other commutators would reduce the intermediate terms significantly.