For the 2D case the construction is simple, because we can find three more points $P_A$, $P_B$, $P_C$ of the ellipse as reflections of $A$, $B$, $C$ about the center $G$ of the ellipse, which is also the centroid of triangle $ABC$.
Moreover, let $A'B'C'$ be the reflections of vertices $ABC$ about the midpoint of the opposite side: triangle $A'B'C'$ has then its sides parallel to the sides of $ABC$, with $A$ being the midpoint of $B'C'$ and so on.
It is a property of the ellipse that the line through the center and the midpoint of a chord, passes through the intersection point of the tangents at the endpoints of the chord. Hence $B'C'$ is tangent to the the Steiner ellipse at $A$, $C'A'$ is tangent at $B$ and $A'B'$ is tangent at $C$.
Moreover, if $M_A$ is the midpoint of chord $BC$, then $GP_A$ is the mean proportional between $GM_A$ and $GA'$.
To find the diameter conjugate to $AP_A$, draw through $G$ the parallel to tangent $B'C'$, to intersect at $D$ and $F$ the other two tangents (see figure). The midpoints $E$ and $H$ of $BP_C$ and $CP_B$ lie on $DF$, hence line $DP_C$ is tangent to the ellipse at $P_C$ and line $FP_B$ is tangent to the ellipse at $P_B$.
The endpoints $IJ$ of the conjugate diameter can then be constructed, because
$GI$ is the mean proportional between $GD$ and $GE$, while $GJ$ is the mean proportional between $GF$ and $GH$.
Once you have a pair of conjugate diameters $AP_A$ and $IJ$, the axes of the ellipse can be constructed as explained here.
