One way to elude the inequality restrictions is with a change of variable so making
#{(a->(sinalpha+1)/2),(b->(sinbeta+1)/2),(c->(singamma+1)/2):}#
the problem now in the variables #alpha, beta, gamma# has only equality restrictions and then we can approach it with the usual lagrange multipliers technique.
#f(alpha,beta,gamma) = 1/8(sinalpha+1)(sinbeta+1)(singamma+1)#
#g(alpha,beta,gamma)=1/4((sinalpha+1)(sinbeta+1)+(sinalpha+1)(singamma+1)+(sinbeta+1)(singamma+1))-1#
with the lagrangian
#L(alpha,beta,gamma,lambda)=f(alpha,beta,gamma)+lambda g(alpha,beta,gamma)#
The stationary point conditions are obtained by solving
#grad L=0#
or
#{(1/8 Cos alpha(5 + 4 lambda + Sin gamma + 2 lambda Sin gamma +
Sin beta (1 + 2 lambda + Sin gamma))=0),(1/8 Cosbeta (5 + 4 lambda + Singamma + 2 lambda Singamma +
Sinalpha(1 + 2 lambda + Singamma))=0),(1/8 Cosgamma(5 + 4 lambda + Sinbeta + 2 lambda Sinbeta +
Sinalpha(1 + 2 lambda + Sinbeta))=0),(1/4 (2 Singamma + Sinbeta (2 + Singamma) +
Sinalpha (2 + Sinbeta + Singamma)-1)=0):}#
Solving this system of equations with the help of Wolfram Alpha we get at
#((a,b,c,f(a,b,c)),(0, 1, 1,2),(1,0,1,2),(1,1,0,2),(1/sqrt[3], 1/sqrt[3], 1/sqrt[3],10/(3 sqrt[3])))#
Here #f(a,b,c) = a + b + c + a b c#
All this trouble could be avoided by observing the problem symmetry, so guessing #a=b=c#