Pencilled inside the back cover of my copy of Knapp's book is a picture that helps me keep a synoptic view of (all?) real form isomorphisms. It is the analog of summarizing the complex isogenies (already explained by Allen) by the statement that the sequence
$$
SO(3,\Bbb C) \to SO(4,\Bbb C) \to SO(5,\Bbb C)\to SO(6,\Bbb C)
$$
(where arrows denote the obvious inclusions) is the $Z_2$ quotient of
$$
Sp(1,\Bbb C) \to SL(2,\Bbb C)^2 \to Sp(2,\Bbb C)\to SL(4,\Bbb C).
$$
What we get for real forms (and let's not forget, complex groups regarded as real) is that **the diagram**

\begin{array}{ccccccccccccc}
O(3,3) &&&& O(4,2) &&&& O(5,1) &&&& O(6)\\\
&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &\\\
&& O(3,2) &&&& O(4,1) &&&& O(5) &&\\\
&\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&&\\\
O(2,2) &&&& O(3,1) &&&& O(4) &&&& \\\
&\nwarrow &&\nearrow &&\nwarrow &&\nearrow && && &\\\
&& O(2,1) &&&& O(3) &&&& &&\\\
&\nearrow &&\nwarrow &&\nearrow &&&& &&&\\\
O(1,1) &&&& O(2) &&&& &&&& \\
\end{array}

**is the same (up to $\pi_0$ and $\pi_1$) as**

\begin{array}{ccccccccccccc}
SL(4,\Bbb R) &&&& SU(2,2) &&&& SU^*(4) &&&& SU(4)\\\
&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &\\\
&& Sp(2,\Bbb R) &&&& Sp(1,1) &&&& Sp(2) &&\\\
&\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&&\\\
SL(2,\Bbb R)^2 &&&& SL(2,\Bbb C) &&&& SU(2)^2 &&&& \\\
&\nwarrow &&\nearrow &&\nwarrow &&\nearrow && && &\\\
&& Sp(1,\Bbb R) &&&& Sp(1) &&&& &&\\\
&\nearrow &&\nwarrow &&\nearrow &&&& &&&\\\
GL(1,\Bbb R) &&&& U(1) &&&& &&&& \\\
\end{array}

Moreover in the latter we could, of course, make any of the substitutions

$$
Sp(1)=SU(2),\quad Sp(1,\Bbb C) = SL(2,\Bbb C),
$$

$$
Sp(1,\Bbb R) = SL(2,\Bbb R) = SU(1,1),
$$

$$
SU^*(4)=SL(2,\Bbb H).
$$

At this point, a good exercise or "proof" is to replace each complex group (resp. real form) by its Dynkin (resp. Satake or Vogan) diagram.