Writing left side of the CASES brackets

One possibility is:

documentclass{article}
usepackage{amsmath}

begin{document}
begin{equation}
    begin{split}
       & E(T,alpha) = 
       & 
begin{matrix}
  textsf{I}    [0.5ex]
  textsf{II}   [0.5ex]
  textsf{III}  [0.5ex]
  textsf{IV}
end{matrix}quad
begin{cases}
    1 & text{$alpha < alpha_{vi}$} 
    textrm{$E_0(T)$} & textrm{$alpha$ = $alpha_{vi}$}
    text{$E_0(T) + (alpha-alpha_{vi})(E_infty(T)-E_{0}(T))$} & text{$alpha_{vi} < alpha < 1$}
    text{$E_infty(T)$} & text{$alpha = 1$}
end{cases}
    end{split}
end{equation}
end{document}

A possibility with {NiceArray} of nicematrix. The output will be correct even if you change the height or the depth or the rows of array.

 documentclass{article}
 usepackage{nicematrix}

 begin{document}

 begin{equation}
 begin{split}
 & E(T,alpha) = 
 &
 begin{NiceArray}{>{color{red}}llc}
 text{I}   & 1 & alpha < alpha_{vi} 
 text{II}  & E_0(T) & alpha$ = $alpha_{vi}
 text{III} & E_0(T) + (alpha-alpha_{vi})(E_infty(T)-E_{0}(T)) & alpha_{vi} < alpha < 1 
 text{IV}  & E_infty(T) & alpha = 1
 CodeAfter
   SubMatrix{{}{1-2}{4-2}{.}
 end{NiceArray}
 end{split}
 label{eq:chile}
 end{equation}

 end{document} 

You need several compilations (because nicematrix uses PGF/Tikz under the hood).

Still another solution, without cases, replaced with blkarray and the bigdelim package:

documentclass{article}
usepackage{amsmath}
usepackage{blkarray}
usepackage{bigdelim}
usepackage[svgnames]{xcolor}

begin{document}

begin{equation}
 begin{split}
 & E(T,alpha) = 
 &
begin{blockarray}{>{color{IndianRed}mathsf{Roman{BAenumi}}}l@{;}c@{}ll}
 & ldelim{{4}{*}& 1 & alpha < alpha_{vi} 
 & & E_0(T) & alpha = alpha_{vi} 
 & & E_0(T) + (alpha-alpha_{vi})(E_infty(T)-E_{0}(T)) & alpha_{vi} < alpha < 1 
 & & E_infty(T) & alpha = 1
end{blockarray}
 end{split}
end{equation}

end{document}