R, 37 bytes
\(m)split(m,pmin(row(m),rev(col(m))))Working on the related challenge led me to this approach. Test harness taken from pajonk's answer.
In an \$m\times n\$ matrix \$A\$, each element \$A_{ij}\$ is in the \$p^\text{th}\$ J-bracket if and only if \$\min(i,n+1-j)=p\$.
R has some odd built-ins that return the matrix \$R=\text{row}(M)\$ where \$R_{ij}=i\$ and similarly for \$C=\text{col}(M)\$. Reversing the column matrix luckily performs the right operation, and we take the parallel minimum of these matrices to obtain a matrix of J-brackets, which split helpfully breaks into groups of the right order.
