Here’s a bit of a laborious symbolic calculation that I messed up by hand with (phy487/oneAtomBasisPhonon.tex)

display = {k1 -> (Subscript[k, 1]), k2 -> (Subscript[k, 2]), k3 -> (Subscript[k, 3]), k4 -> (Subscript[k, 4]), s1 -> (Subscript[s, 1]), s2 -> (Subscript[s, 2]), s3 -> (Subscript[s, 3]), s4 -> (Subscript[s, 4]), sign -> "±"} ;

Clear[aa, ei, lambda1, lambda2, e1, e2, f1, f2] 
aa = (2/m){ {2k1 s1 + k3 s3 + k4 s4, 1(k3 s3 - k4 s4)}, {1(k3 s3 - k4 s4), 2 k2 s2 + k3 s3 + k4 s4}} ;

Row[ {"A = ",(2/m // TraditionalForm), (aa m /2/. display// Factor// TraditionalForm// MatrixForm )}]

ei = Eigensystem[ aa ] ;
{lambda1, lambda2} = ei[[1]] ;
{e1, e2} = ei[[2]] ;
Row[{(SubPlus[OverVector["e"]])," = ", e2 (k3 s3 - k4 s4)/. display // FullSimplify // Factor // MatrixForm // TraditionalForm }]
Row[{(SubMinus[OverVector["e"]])," = ", e1 (k3 s3 - k4 s4)/. display // FullSimplify // Factor // MatrixForm // TraditionalForm }]
Row[{"\!\(\*SubscriptBox[\(λ\), \(\(+\)\(\\ \)\)]\)= ", (2/m  // TraditionalForm), " \[Cross] (",(lambda2 m /2/. display // FullSimplify // Factor // TraditionalForm ), ")"}]
Row[{"\!\(\*SubscriptBox[\(λ\), \(\(-\)\(\\ \)\)]\)= ", (2/m  // TraditionalForm)," \[Cross] "(lambda1 m /2/. display // FullSimplify // Factor // MatrixForm // TraditionalForm) }]

(*ei  /. display // FullSimplify // Factor // TraditionalForm*)

Clear[f, f1, f2]
f=  { k3 s3 - k4 s4, k2 s2 - k1 s1  + # Sqrt[ (k1 s1 - k2 s2)^2  + (k3 s3 - k4 s4)^2]} & ;
(*Table[(f[i] /. display // MatrixForm // TraditionalForm), {i, 1, -1, -2}] ;*)
f1 = f[1] ;
f2 = f[-1] ;

Row[ {"(SubPlus[f]) = ", (f1 /. display // MatrixForm // TraditionalForm)}]
Row[ {"(SubMinus[f]) = ", (f2 /. display // MatrixForm // TraditionalForm)}]
(aa . f1 - lambda2 f1 )/. display // ExpandAll // MatrixForm // TraditionalForm
(aa . f2 - lambda1 f2 )/. display // ExpandAll // MatrixForm // TraditionalForm

Now in a rotated (by π/4) coordinate system.

Clear[aa, ei, lambda1, lambda2, e1, e2] 
aa = (2/m){ {k1 s1 + k2 s2 + 2 k3 s3 , 1(k2 s2 - k1 s1)}, {1(k2 s2 - k1 s1), k1 s1 + k2 s2 + 2 k4 s4 }} ;

Row[ {"A = ",(2/m // TraditionalForm), (aa m /2/. display// Factor// TraditionalForm// MatrixForm )}]

ei = Eigensystem[ aa ] ;
{lambda1, lambda2} = ei[[1]] ;
{e1, e2} = ei[[2]] ;
Row[{(SubPlus[OverVector["e"]])," = ", e2 (k1 s1 - k2 s2)/. display // FullSimplify // Factor // MatrixForm // TraditionalForm }]
Row[{(SubMinus[OverVector["e"]])," = ", e1 (k1 s1 - k2 s2)/. display // FullSimplify // Factor // MatrixForm // TraditionalForm }]
Row[{"\!\(\*SubscriptBox[\(λ\), \(\(+\)\(\\ \)\)]\)= ", (2/m  // TraditionalForm), " \[Cross] (",(lambda2 m /2/. display // FullSimplify // Factor // TraditionalForm ), ")"}]
Row[{"\!\(\*SubscriptBox[\(λ\), \(\(-\)\(\\ \)\)]\)= ", (2/m  // TraditionalForm)," \[Cross] "(lambda1 m /2/. display // FullSimplify // Factor // MatrixForm // TraditionalForm) }]

(*ei  /. display // FullSimplify // Factor // TraditionalForm*)

Clear[f, f1, f2]
f=  { k2 s2 - k1 s1, k4 s4 - k3 s3  + # Sqrt[ (k1 s1 - k2 s2)^2  + (k3 s3 - k4 s4)^2]} & ;
(*Table[(f[i] /. display // MatrixForm // TraditionalForm), {i, 1, -1, -2}] ;*)
f1 = f[1] ;
f2 = f[-1] ;

Row[ {"(SubPlus[f]) = ", (f1 /. display // MatrixForm // TraditionalForm)}]
Row[ {"(SubMinus[f]) = ", (f2 /. display // MatrixForm // TraditionalForm)}]
(aa . f1 - lambda2 f1 )/. display // ExpandAll // MatrixForm // TraditionalForm
(aa . f2 - lambda1 f2 )/. display // ExpandAll // MatrixForm // TraditionalForm