notes: add executing commands section

NOTES.md

@@ -22,6 +22,8 @@ of filenames and commands and arguments.
 Performs any necessary _redirections_ and removes the **redirection operators**
 and their operands from the argument list.
 
+Executes the command (see _Command Execution_);
+
 TODO: add missing operations
 
 ### Quoting Rules
@@ -367,8 +369,61 @@ bash-5.1$ echo "hello world" > $nonexist
 bash: $nonexist: ambiguous redirect
 ```
 
-## Subshell
+### Executing Commands
+cf. 3.7 Executing Commands
 
+#### Simple Command Execution
+cf. 3.7.1 Simple Command Expansion
+
+When a simple command is executed, the shell performs the following
+expansions, assignments, and redirections, from left to right, in the
+following order.
+
+  1. The words that the parser has marked as redirections are saved for later
+     processing.
+
+  2. The words that are not redirections are expanded (see _Shell Expansions_).
+     If any words remain after expansion, the first word is taken to be the name
+     of the command and the remaining words are the arguments.
+
+  3. Redirections are performed as described above (see _Redirections_).
+
+If no command name results, redirections are performed, but do not affect the
+current shell environment.  A redirection error causes the command to exit with
+a non-zero status.
+
+If there is a command name left after expansion, execution proceeds as described
+below.  Otherwise, the command exits with a status of zero.
+
+##### Command Search and Execution
+cf. 3.7.2 Command Search and Execution
+
+After a command has been split into words, if it results in a simple command and
+an optional list of arguments, the following actions are taken.
+
+  1. The shell searches for it in the list of shell builtins.  If a match is
+     found, that builtin is invoked.
+
+  2. If the name is not a builtin, and contains no slashes, Bash searches each
+     element of '$PATH' for a directory containing an executable file by that
+     name.  If the search is unsuccessful, the shell prints an error message and
+     returns an exit status of 127.
+
+  3. If the search is successful, or if the command name contains one or more
+     slashes, the shell executes the named program in a separate execution
+     environment.  Argument 0 is set to the name given, and the remaining
+     arguments to the command are set to the arguments supplied, if any.
+
+  4. If this execution fails because the file is not in executable format, and
+     the file is not a directory, it is assumed to be a "shell script" and the
+     shell executes it as described in _Shell Scripts_.
+     TODO: check if we need to implement the _Shell Scripts_ behaviour
+
+  5. The shell waits for the command to complete and collects its exit status.
+
+TODO remove $@ $* mentions as we don't have to implement that
+
+#### Subshell
 cf. 3.7.3 Command Execution Environment
 
 The shell has an execution environment, which consists of the following: