This paper is devoted to a general solvability of a multi-dimensional backward stochastic differential equation (BSDE) of a diagonally quadratic generator $g(t,y,z)$, by relaxing the assumptions of \citet{HuTang2016SPA} on the generator and terminal value. More precisely, the generator $g(t,y,z)$ can have more general growth and continuity in $y$ in the local solution; while in the global solution, the generator $g(t,y,z)$ can have a skew sub-quadratic but in addition "strictly and diagonally" quadratic growth in the second unknown variable $z$, or the terminal value can be unbounded but the generator $g(t,y,z)$ is "diagonally dependent" on the second unknown variable $z$ (i.e., the $i$-th component $g^i$ of the generator $g$ only depends on the $i$-th row $z^i$ of the variable $z$ for each $i=1,\cdots,n$ ). Three new results are established on the local and global solutions when the terminal value is bounded and the generator $g$ is subject to some general assumptions. When the terminal value is unbounded but is of exponential moments of arbitrary order, an existence and uniqueness result is given under the assumptions that the generator $g(t,y,z)$ is Lipschitz continuous in the first unknown variable $y$, and varies with the second unknown variable $z$ in a "diagonal" , "component-wisely convex or concave", and "quadratically growing" way, which seems to be the first general solvability of systems of quadratic BSDEs with unbounded terminal values. This generalizes and strengthens some existing results via some new ideas.