Lorentz transformation matrix for all 3 spatial axes

The lorentz transformation matrix (for all 3 spatial axes, not just a single dimension boost) appears to be commonly defined as the following:
$$
begin{bmatrix}
gamma &-gamma v_x/c &-gamma v_y/c &-gamma v_z/c
-gamma v_x/c&1+(gamma-1)dfrac{v_x^2} {v^2}& (gamma-1)dfrac{v_x v_y}{v^2}& (gamma-1)dfrac{v_x v_z}{v^2}
-gamma v_y/c& (gamma-1)dfrac{v_y v_x}{v^2}&1+(gamma-1)dfrac{v_y^2} {v^2}& (gamma-1)dfrac{v_y v_z}{v^2}
-gamma v_z/c& (gamma-1)dfrac{v_z v_x}{v^2}& (gamma-1)dfrac{v_z v_y}{v^2}&1+(gamma-1)dfrac{v_z^2} {v^2}
end{bmatrix}
$$
I tried to derive it myself by combining the matrices for the individual boost directions and making $v=|vec{v}|$and ended up at
$$
begin{bmatrix} ct’ x’ y’ z’ end{bmatrix} = begin{bmatrix} gamma & -beta_xgamma& -beta_ygamma & -beta_zgamma -frac{beta_y} {gamma_{v_x}} & frac{1}{gamma_{v_x}} & 0 & 0 -frac{beta_y}{gamma_{v_y}} & 0 & frac{1}{gamma_{v_y}} & 0-frac{beta_z}{gamma_{v_z}} & 0 & 0 & frac{1}{gamma_{v_z}} end{bmatrix} begin{bmatrix} ct x y z end{bmatrix}
$$
Where $gamma = displaystylefrac{1}{sqrt{1-displaystylefrac{|vec{v}|^2}{c^2}}} $ and
$gamma_{v_x} = displaystylefrac{1}{sqrt{1-displaystylefrac{v_x^2}{c^2}}}$

2 questions. Where do the bottom right 9 terms come from in the common definition and why is the top $gamma$ and not $frac{1}{gamma}$ given that $l′=frac{l}{gamma}$ but $t′=tgamma$